Gray Molasses Cooling of Lithium-6 Towards a Degenerate Fermi Gas

Department of Physics and Astronomy

University of Heidelberg

Master thesis in Physics

submitted by

Manuel Gerken

born in Frankfurt am Main

2016

Gray Molasses Cooling of Lithium-6 Towards a

Degenerate Fermi Gas

This Master thesis has been carried out by Manuel Gerken

at the

Physikalisches Institut Heidelberg

under the supervision of

Prof. Dr. Matthias Weidemüller iv Kühlen mit grauer Melasse zur Realisierung eines entarteten Fermi Gases

Die vorliegende Arbeit beschreibt das Design, die Implementierung und die Charakteri- sierung einer Kühlung in grauer Melasse auf der D1 Linie von Lithium-6 Atomen. Durch diese zusätzliche Laser-Kühlung erreichen wir sub-Doppler Temperaturen für eine an- fänglich magneto-optisch gefangenes Gas und erhöhen seine Phasenraumdichte um einen Faktor von etwa 10. Kühlen mit grauer Melasse kombiniert die geschwindigkeitsabhängige Besetzung eines kohärenten Dunkelzustands mit einem sisyphusartigen Kühlprozess auf einer ortsabhän- gigen Energieverschiebung. Wir präsentieren Berechnungen und Analysen von bekleide- ten Energiezuständen von Lithium-6 in ein- und dreidimensionalen Polarisationsgradien- tenfeldern. Dieser erzeugen die ortsabhängige Energieverschiebung für die Kühlung mit grauer Melasse. Wir beschreiben die Planung und den Aufbau des Experiments, in dem 3.2×107 Atome innerhalb von 1 ms von 240 µK auf 42 µK herabgekühlt werden. Dies ent- spricht einer Einfangeffizienz von 80% gegenüber der anfangs in der magneto-optischen Falle hergestellten Probe. Wir messen die Auswirkungen von Frequenzverstimmung, Dau- er, Magnetfeld und Lichtintensitäten auf den Kühlprozess um optimale Parameter zu er- halten. Wir diskutieren, wie die erhöhte Phasenraumdichte die Übertragungseffizienz in die optische Dipolfalle verbessert. Die verbesserten Ausgangsbedingungen für evaporati- ves Kühlen ermöglichen es uns, ein stark entartetes Fermigas herzustellen, der Grundbau- stein für die Generierung einer suprafluiden Bose-Fermi-Mischung und die Untersuchung von Fermi Polaronen.

Gray Molasses Cooling of Lithium-6 Towards a Degenerate Fermi Gas

This thesis presents the design, implementation and characterization of gray molasses cooling on the D1 line of Lithium-6 atoms. With this approach we reach sub-Doppler temperatures for an initially magneto-optically trapped gas and increase its phase space density by a factor of approximately 10. Gray molasses cooling consists of a velocity-selective coherent population trapping of atoms at zero velocity in a dark state that works in combination with a Sisyphus-like cooling process in a spatially dependent light-shift. We present calculations and analysis of dressed state energies of Lithium-6 in one and three dimensional polarization gradient fields, which generate the spatially dependent energy shift of the gray molasses cooling. Design and setup for the experiment are described, in which 3.2 × 107 atoms are cooled from 240 µK to 42 µK in 1 ms. The atom number represents an 80% capture efficiency of the initial sample prepared in the magneto-optical trap. We measure and optimize the impact of frequency detuning, duration, magnetic field and beam intensities onto the cooling process. We discuss how the increased phase space density will improve the transfer efficiency into the optical dipole trap. The enhanced starting conditions for forced evaporative cooling will enable us to achieve a highly degenerate Fermi gas, the keystone to a double superfluid Bose-Fermi mixture and the investigation of Fermi polarons.

Contents

1 Introduction 1

2 Concept of gray molasses cooling 5 2.1 Principle of gray molasses cooling ...... 5 2.2 Λ configuration and coherent dark state ...... 7 2.3 Polarization gradient fields ...... 9 2.4 Gray molasses of Lithium-6 ...... 12 2.5 Calculations for Lithium-6 in gray molasses ...... 15 2.5.1 Description of light polarization ...... 15 2.5.2 Hamiltonian describing gray molasses ...... 17 2.5.3 Discussion of gray molasses in 1D Lin⊥Lin field ...... 20 2.5.4 Discussion of gray molasses in a 3D σ+-σ− field ...... 24

3 Implementation of gray molasses cooling of Lithium-6 27 3.1 Setup of the gray molasses ...... 27 3.1.1 Laser system and stabilization ...... 28 3.1.2 Optical setup for light preparation ...... 30 3.1.3 Optical setup at experimental chamber ...... 32 3.2 Characterization of gray molasses cooling ...... 33 3.2.1 Experimental sequence ...... 33 3.2.2 Measurements of gray molasses cooling ...... 35

4 Conclusion and further developments 45

Bibliography 53

Acknowledgment 63

vii

1 Introduction

Ultracold atomic gases yield an advantageous environment for the probing and simulation of few and many-body physics (Bloch et al., 2008). The features of controlling internal degrees of freedom, external motional states, the density of the gas, the potential shape and even the atomic interactions, of a cloud of atoms or a mixture of atoms, makes the field feasible for the investigation of a wide range of physical questions. One example is the simulation of magnetic spins in a condensed matter system where the interaction between the spins is of a long- range nature and spatially anisotropic. Such a system can be simulated with an ultracold gas of polar molecules where the strength of the electric dipole-dipole interaction can be controlled to uttermost accuracy (Pupillo et al., 2008; Quéméner and Julienne, 2012). Another example are Fermi impurities interacting with a degenerate Bose environment. This system simulates electrons moving in a lattice of positively charged atoms in a crystal. The interaction between the electron and the environment leads to a quasi-particle called "Fröhlich-polaron" with a different mass and energy compared to the bare electron (Fröhlich, 1954; Tempere et al., 2009). The last example is the generation of a double superfluid phase of a Bose- Fermi mixture. The historical approach to achieve this phase using Helium-4 and Helium-3 reaches a limit when decreasing the temperature. The density of Helium-4 drops to zero before Helium-3 reaches superfluidity (Edwards, 2013). Such a system can be simulated using ultracolt atomic gases. All of these three examples can be studied in an ultracold Bose-Fermi mixture of Cesium-133 and Lithium-6 under favorable conditions. A polar molecule of Cesium- 133 and Lithium-6 holds the strongest electric dipole moment in the singlet ground state of all alkali combinations leading to enhanced interaction effects (Aymar and Dulieu, 2005). The high mass ratio of Cesium-133 and Lithium-6 of 22.1:1 leads to simplifications in the theoretic description for Fermi polarons and double superfluidity. Theories for static polarons, in the limit of infinitely heavy impurities, can directly be tested in the case of Cesium-133 impurities immersed in a Lithium-6

1 Chapter 1. Introduction

Fermi sea. The mass ratio is predicted to change the interaction energy between the two superfluids (Zhang et al., 2014). The study of all these fundamental properties of a strongly interaction Bose-Fermi gas will benefit from a favorable scattering resonances between Lithium-6 and Cesium-133, leading to controllable intra and inter species interactions (Tung et al., 2013; Repp et al., 2013; Pires et al., 2014; Ulmanis et al., 2015). A long journey of progress in technical and physical understandings of 40 years lead to the point where we can address these questions in ultracold atomic systems. The proposal by Hänsch and Schawlow (1975) to cool a vapor of atoms by intense quasi-monochromatic laser light in 1975 was the beginning of laser cooling, a milestone in ultracold atomic physics. In the following years laser cooling and trapping methods were developed and awarded with the Nobel Prize in 1997 (Phillips, 1998; Chu, 1998; Cohen-Tannoudji, 1998). The realization of Bose-Einstein condensation (BEC) in diluted atomic gases in 1995 (Davis et al., 1995; Bradley et al., 1995; An- derson et al., 1995) was the first experimental generation of a degenerate quantum gas and was honored with the Nobel prize in 2001 (Cornell and Wieman, 2002; Ketterle, 2002). Due to the ultralow temperature, a dilute Bose gas with a phase space density of approximately one can be produced. In such a gas a macroscopic part of the ensemble occupies the lowest energy state of the system. This leads to a macroscopic quantum wavefunction. A few years after the generation of BEC the first degenerate Fermi gas was realized by laser cooled atoms (DeMarco and Jin, 1999; Schreck et al., 2001; Truscott et al., 2001; Hadzibabic et al., 2002). The discovery of magnetic Feshbach resonances in 1998 marked another milestone in the field of ultracold atomic physics (Inouye et al., 1998; Courteille et al., 1998). Such scattering resonances allow to control and tune the complete scattering process at ultracold temperatures, where it is described by the s-wave scattering length a. This ability of tuning the interaction between two different atoms or two spin components of the same atomic species opened the door for the observation of many new effects. The first realization of a BEC out of weakly bound molecules (a > 0), formed from two fermionic components, was achieved in 2003 (Jochim et al., 2003; Zwierlein et al., 2003; Greiner et al., 2003). For weakly attractive interactions (a < 0) pairing mechanism of a two component Fermi gas could be investigated (Chin et al., 2004; Schunck et al., 2008). This mechanism describes low temperature superconductivity and was predicted by the Bardeen-Cooper-Schrieffer theory (BCS) (Bardeen et al., 1957).

2 Ultracold gases with tunable interactions have been applied to address polaron physics and double superfluidity. Attractive and repulsive Fermi polarons have been detected for impurities in a highly degenerate, single-species fermionic sample by changing the internal state of the impurity from a non-interacting into an interacting state while tuning the interactions between bath and impurity using Feshbach resonances (Schirotzek et al., 2009; Kohstall et al., 2012; Koschorreck et al., 2012). In 2016 ultrafast dynamics of impurities immersed in a single species Fermi sea were investigated (Cetina et al., 2016). The amplitude damping of the coherent dynamics was directly connected to the degeneracy of the bath. A high degeneracy is thus necessary for precise observation of the collective response. Only in 2016 the first observations of attractive and repulsive Bose polarons in a BEC were reported (Jørgensen et al., 2016; Hu et al., 2016). In Bose-Fermi mixtures the first double superfluid was realized by Ferrier-Barbut et al. (2014) in a mixture of Lithium-7 and Lithium-6 and by Roy et al. (2016) in a mixture of Ytterbium-174 and Lithium-6. Coherent energy exchange between both species was detected by driving center-of-mass oscillations. Yao et al. (2016) found visual evidence for double superfluidity in a mixture of Potassium-41 and Lithium-6 by rotating the gas and therefore creating simultaneously existing vortex lattices. Neither of these effects have been realized in a mixture of Cesium-133 and Lithium-6. While a BEC of Cesium-133 (Weber et al., 2003; Kraemer et al., 2004; Pires et al., 2014) and superfluidity of Lithium-6 are nowadays routinely realized in many groups (Zwierlein et al., 2006b; Sidorenkov et al., 2013; Ferrier-Barbut et al., 2014; Yao et al., 2016; Roy et al., 2016), it has so far not been possible to reach superfluidity of either lithium or cesium in the presence of the other species. In our group we are able to produce degenerate Lithium-6 and Cesium-133 samples separately. The Cesium-133 BEC sample contains 5×104 atoms with a condensate fraction of approximately 70% at a magnetic field of 22.8 G (Pires, 2014). For Lithium-6 we achieve 5×104 atoms with a condensate fraction of 70% at a magnetic field of 690 G (Repp, 2013). However superfluid Lithium-6 on the BCS side (a < 0) has not been realized in our experiment. The sample temperature to achieve superfluidity for a two component Fermi gas at negative scattering length for the specif harmonic trapping potential in our experiment is given by approximately

T . 10−10N 1/3K. (1.1)

Here T is the sample temperature and N is the number of atoms. In order to reach

3 Chapter 1. Introduction superﬂuidity, either the temperature has to be reduced or the number of atoms has to be increased. The latter can be achieved by enhancing phase space density of the Lithium-6 cloud before transferring it into the optical dipole trap to improve starting conditions for forced evaporative cooling (Ketterle and van Druten, 1996).

Due to the unresolved hyperfine splitting of the D2 line of Lithium-6 the sub- Doppler cooling process, working in the magneto-optical trap (MOT) of, for example sodium (Lett et al., 1988) and other species, does not work here. This leads to a MOT temperature that is limited by the Doppler temperature which is around 140 µK for Lithium-6. Two approaches have been used to reach lower temperatures. Narrow-line cooling (Duarte et al., 2011; McKay et al., 2011) applied on 2 2 the 2 S1/2 → 3 P3/2 transition of Lithium-6 at 323 nm, provides temperatures of ≈ 60 µK, increasing the phase space density of the MOT by one order of magnitude. Due to the narrow linewidth of the uv transition of Γ = 2π × 754 kHz the Doppler temperature decreases to 18 µK. Another method is the one we present here. Gray molasses cooling on the D1 line is a process closely related to Sisyphus cooling. It is based on the concept of velocity selective coherent population trapping (VSCPT) (G. Grynberg, 1994; Weidemüller et al., 1994). Using this method temperatures down to 60 µK, 40 µK, 20 µK and 6 µK have been reached for Lithium-7, Lithium-6, Potassium-40 and Potassium-39, respectively (Grier et al., 2013; Burchianti et al., 2014; Rio Fernandes et al., 2012; Salomon et al., 2013). In this way a temperature decrease of approximately one order of magnitude can be realized for Lithium-6 while maintaining the MOT atom number and sample density. This thesis covers calculations, design, implementation, characterization and future plans of gray molasses cooling in the existing experimental setup. In chapter 2 we give an overview over the working principles of gray molasses cooling by first presenting a simplified cooling scheme in a three-level Λ system and later extending the model to the case of Lithium-6. We present calculations for Lithium-6 in a one and three dimensional gray molasses field with two different polarization configurations. Chapter 3 presents the experimental setup, implementation as well as the characterization of the gray molasses cooling. In the final chapter we summarize the findings and give an outlook on experiment prospects coming into reach by gray molasses cooling.

4 2 Concept of gray molasses cooling

In this chapter we discuss the basic principles of gray molasses cooling on the D1 line of Lithium-6. At first we give a brief overview over the cooling process in a gray molasses in section 2.1. In section 2.2 we take a closer look at a 3-level Λ configuration and map out three basic points to explain the cooling process. We examine the effect of a light polarization gradient field onto the system and discuss two possible field configurations. Section 2.4 introduces Lithium-6 and discusses the effect of the gray molasses. In section 2.5 we present calculations for Lithium-6 in two different polarization gradient field configurations in one and three dimensions, which gives us a deeper understanding of the cooling mechanism and its dependence on the Raman detuning, the phase difference between two frequency components and the magnetic field.

2.1 Principle of gray molasses cooling

The main idea of gray molasses cooling can be described as follows: An atom propagates in a blue detuned light field with spatially dependent polarization. The field couples atomic states, such that the ground state manifold is divided into bright states |ψB⟩ and dark states |ψD⟩. The dark states experience no energy shift due to the light field. The bright states however experience a spatially varying light shift induced by the polarization gradient of the field. Figure 2.1 shows a schematic picture of bright state |ψB⟩ energy and dark state |ψD⟩ energy along the z axis in the light field. Now we consider an atom with velocity v propagating along the light field in positive z direction. In the bright state, the atom will exchange kinetic and potential energy when climbing a potential hill. At the point of highest light shift it will have the lowest velocity. The probability of the atom to absorb a photon is proportional to the light shift, thus it is more probable for an atom to get excited at the top of the potential hill than in the potential valley. After absorbing a photon, the atom will either decay back into the bright state where it will again absorb a

5 Chapter 2. ConceptD. Rio of gray Fernandes molasseset al. cooling

E 훿 F'=7/2 2 P1/2 훿 2 155.3 MHz

F'=9/2

Cooling ψB 770.1 nm 0 Repumping

ψD

z F=7/2 λ/4 λ/2 3λ/4

0 2 Fig. 1: Gray Molasses scheme. On a F → F = F or S1/2 1285.8 MHz Figure 2.1: A positiveF → F detuned0 = F − 1 optical optical transition transition with in a spatialpositive dependent detuning, polarization fieldthe splits ground the ground state splits states into into a dark a bright and a bright state | manifoldψB⟩ and a dark state |ψD⟩ withmanifold. positive Kinetic energy, energy shown as is transfered|ψDi and |ψ toBi potentialrespectively. energy when an F=9/2 In the presence of a polarization gradient, the bright state atom climbs the potential hill of the bright state. When pumpedFig. 2: from Level scheme for the D transition of 40K and energy is spatially modulated. Like in Sisyphus cooling, 1 the bright to the dark state the atom looses energy. Atoms transfertransitions from used for gray molasses cooling. The laser de- energy is lost when an atom in |ψ i climbs a potential the dark to the bright state due to velocityB selective coupling.tuning Taken from the cooling/repumping transitions is δ and hill before being pumped back into the dark state |ψ i. D the detuning from the off-resonant excited hyperfine state fromMotional (Rio Fernandes coupling et between al., 2012|ψ).i and |ψ i occurs prefer- D B F 0 = 9/2 is δ (see text). entially at the potential minima. 2 photon, or it will decay into the dark state. The decay into the dark state causes the atom to loseon energy the F on→ theF 0 order= F ( ofF the→ F bright0 = F state− 1) energy optical shift. tran- As mentioned above, two mechanisms can lead to the sition. For any polarization of the local electromagnetic departure from the dark state. The first one is the mo- An atom populatingfield, the the ground dark state state manifold can transfer possesses to the one bright (two) dark statetional via velocity coupling Vmot due to the spatial variations of the selective coupling.states The which probability are not optically of the process coupled is to proportional the excited state to thedark velocity state internalof wave-function induced by polarization by the incident light [12,16]. When the laser is detuned to and intensity gradients. The second one is the dipolar the atom and inversely proportional to the light shift of the bright state (Dalibard the blue side of the resonance, the ground state manifold coupling Voff via off-resonant excited hyperfine states. A and Cohen-Tannoudjisplits, into1989 dark). This states process which are leads not to affected velocity by selective light and coherentrough estimate pop- shows that Vmot ' ~kv, where v is the bright states which are light-shifted to positive energy by velocity of the atom and k the wave-vector of the cool- ulation trapping (VSCPT) (Arimondo, 1992). Any atom with a velocity other than −1 an amount which depends on the actual polarization and ing light, while Voff ' ~Γ (Γ/δ2) I/Isat, where Γ is the zero will take partintensity in the of cooling the laser cycle field until (see fig. reaching 1). zero velocity. Atomslifetime at of rest the excited state, I the light intensity, Isat the will accumulate inWhen the dark the state. atom is The in a process bright state, is therefore it climbs not up thelimited hill bysaturation the single intensity and δ2 the detuning to off-resonant ex- of the optical potential before being pumped back to the cited state. Comparing the two couplings, we see that the photon recoil energydark state which near is the the top limit of of the Sisyphus hill. The cooling kinetic energy (Castin of etmotional al., 1991 coupling). is significant in the high velocity regime Thus temperaturesthe below atom is the thus single reduced photon by an recoil amount temperature of the order of the v & Γ/k (Γ/δ2) I/Isat. In our case, the off-resonant level 0 height of the optical potential barrier. The cooling cycle F = 9/2 (see fig. 2) is detuned by δ2 = 155.3 MHz+δ from 2 2 0 ~2 2 the cooling transition | S1/2,F = 9/2i → | P1/2,F = is completed near the potentialk minimap by a combination T = , 7/2i. For(2.1)I ' I , motional coupling dominates for of motional couplingrecoil and optical2mk excitation to off-resonant sat hyperfine states. B T & 50 µK, meaning that both processes are expected to 40 be present in our experiments. In general, the transition where k is the photonWe implement wave number 3D gray and molassesm is cooling the atom in mass,K on the can be achieved p rate between |ψ i and |ψ i induced by motional coupling D1 transition (see fig. 2). In alkali atoms, the P1/2 excited D B (Esslinger et al.,level1996 manifold). However, has only off-resonant two hyperfine coupling states, and which imperfections are Vmot and in the the off-resonant coupling Voff are both maximal when the distance between the dark and bright manifolds better resolved than their P3/2 counterparts. These facts allow for less off-resonant excitation and a good control of is smallest, which favors transitions near the bottom of the cooling mechanism. A first laser beam (cooling beam) the wells of the optical lattice. 6 2 2 0 40 is tuned to the | S1/2,F = 9/2i → | P1/2,F = 7/2i In K, the simplified discussion presented so far must transition with a detuning δ > 0. A second laser beam be generalized to the case involving many hyperfine states 2 (repumping beam) is tuned to the | S1/2,F = 7/2i → (10 + 8). However, the essential picture remains valid. In- 2 0 | P1/2,F = 7/2i transition with the same detuning δ. deed, by numerically solving the optical Bloch equations

p-2 2.2. Λ conﬁguration and coherent dark state polarization ﬁeld can disturb the process and limit the achievable temperature. In some systems temperatures above the recoil limit but still an order of magnitude beneath the Doppler limit can be reached (Grier et al., 2013).

2.2 Λ conﬁguration and coherent dark state

To describe the emergence of dark and bright states and velocity selective coupling in a simple picture we here introduce the Λ three-level system. The Λ three-level model is shown in ﬁgure 2.2. It describes a system with two ground states |1⟩ and |2⟩ and one excited state |3⟩, with energies ϵ1, ϵ2 and ϵ3, respectively. States | ⟩ | ⟩ 1 and√3 are coupled via an oscillator with frequency ω1 and Rabi frequency Ω1 = Γ I1/2Isat, where the frequency is detuned by δ1 from resonance, Γ is the linewidth of state |3⟩ and I1 the intensity of the light ﬁeld. Isat is the saturation intensity (Weidemüller and Zimmermann, 2003) of the atomic transition with

1 Γ~ω3 I = 1 , (2.2) sat 12 π2c2 | ⟩ | ⟩ where c is the speed of light. States 2 and 3 are coupled by an oscillator√ with frequency ω2, detuned by δ2 from resonance and Rabi frequency Ω2 = Γ I2/2Isat.

The relative detuning of ω1 and ω2 is ∆ = δ1 − δ2. Diagonalizing the Hamiltonian of the system will give us the dressed states, describing the dark and the bright state. Under several conditions the system simpliﬁes dramatically. First we assume no diﬀerence in energy of state |1⟩ and |2⟩ meaning ϵ1 = ϵ2. Furthermore we assume the so called Raman condition ∆ = 0.

We thus have ω1 = ω2. The Hamiltonian of this complete system H can be divided into the Hamiltonian without light interaction H0 and the coupling Hamiltonian V . In the rotating frame the Hamiltonian H can be written as:    0 0 Ω1  ~   H = H + V =  0 0 Ω  , (2.3) 0 2  2  Ω1 Ω2 −2δ1 where H0 = −~δ1 |3⟩⟨3| is the Hamiltonian of the system without light coupling and −~δ1 is the energy of the excited state. The coupling Hamiltonian is given by

~Ω ~Ω V = 1 |1⟩⟨3| + 2 |2⟩⟨3| + c.c. (2.4) 2 2

7 Chapter 2. Concept of gray molasses cooling

Figure 2.2: Three-level scheme coupled by two light ﬁelds with Rabi frequencies Ω1 and Ω2 detuned from resonance by δ1 and δ2, respectively. Adapted from (Grier et al., 2013).

It describes the coupling of states |1⟩, |2⟩ and |3⟩ by the light ﬁelds ω1 and ω2 where

ω1 = ω2 but Ω1 ≠ Ω2. After diagonalizing the Hamiltonian, we get two new eigenstates in the dressed state picture √ 1 |ψD⟩ = (Ω1 |1⟩ − Ω2 |2⟩), (2.5) 2 2 Ω1 + Ω2

√ 1 |ψB⟩ = (Ω1 |1⟩ + Ω2 |2⟩). (2.6) 2 2 Ω1 + Ω2

In order to determine whether the states couple to the light ﬁeld or experience a light shift, we examine the eﬀect of the coupling Hamiltonian on them. For state

|ψD⟩ we get V |ψD⟩ = 0. This means that state |ψD⟩ does not couple to the light | ⟩ ﬁeld and does not experience a light√ shift. Consequently we call ψD the dark state. | ⟩ ⟨ | | ⟩ ~ 2 2 | ⟩ For ψB , we get 3 V ψB = 2 Ω1 + Ω2. This shows that state ψB couples to (Ω2+Ω2) the excited state |3⟩ via the light ﬁeld. It experiences a light shift 1 2 and δ1 we therefore call it the bright state. For a positive (blue) detuning δ1 > 0 the light shift is positive, while for a negative (red) detuning δ1 < 0 the light shift is negative. Since |ψB⟩ and |ψD⟩ are orthogonal eigenstates in the basis of H, they do not couple, so that ⟨ψB| H |ψD⟩ = 0.

We now introduce velocity dependent coupling between |ψD⟩ and |ψB⟩. This was described in (Papoﬀ et al., 1992) for the Λ three-level scheme. We will here brieﬂy summarize the most relevant results. For this we include the kinetic energy term in H pˆ2 the full Hamiltonian = 2m + H, where H is the Hamiltonian from equation 2.3 and pˆ is the momentum operator.The coupling matrix element between the dark

8 2.3. Polarization gradient ﬁelds and the bright state yield

⟨ | H | ⟩ −~ 2Ω1Ω2 kp ψB ψD = 2 2 , (2.7) Ω1 + Ω2 m where p is the momentum of the atom and k is the wave number of the transitions. This shows a coupling depending on the velocity of the atom v = p/m. Papoﬀ et al. (1992) calculate in ﬁrst order perturbation theory the coupling probability ( ) 2 Ω1Ω2 p δ1 P| ⟩→| ⟩ = 2 k δ , (2.8) ΨD ΨB 2 2 ~ 1 2 2 Ω1 + Ω2 (Ω1 + Ω2) which describes the velocity selective coupling of an atom with momentum p from the dark state |ψD⟩ to the bright state |ψB⟩. Therefore the transition probability

δ1 is proportional to the velocity square. The last factor 2 2 describes the inverse (Ω1+Ω2) of the light shift of the bright state. δ1 describes the detuning from resonance, thus the coupling will be strongest at small light shifts and high velocity. This shows that all the slow atoms accumulate in the dark state |ψD⟩. The fast atoms are involved in the cooling cycle until the velocity is zero such that the motional coupling vanishes. With this model we can describe the following: An atom with high kinetic energy in the dark state |ΨD⟩ will couple to the bright |ψB⟩ state at a point close to the potential minimum of the bright state. Then moving in the light ﬁeld it will climb the potential hill. After an excitation by photon absorption, it will decay back into the ground state manifold. If the atom decays into the bright state the process is repeated. If it decays into the dark state the atom looses kinetic energy on the order of the bright state light shift and is thus cooled.

2.3 Polarization gradient ﬁelds

To understand the origin of a spatial dependent bright state energy shift we will here discus the inﬂuence of a polarization gradient light ﬁeld on an atom. For this we consider a system with a ground state of total angular momentum F = 1/2 which we denote |F = 1/2⟩ and an excited state of total angular momentum F ′ = 3/2 which we denote |F ′ = 3/2⟩. The states then consist of degenerate sub-

{− 1 1 } ′ states described by the magnetic quantum numbers mF = 2 , 2 and mF = {− 3 − 1 1 3 } 2 , 2 , 2 , 2 which are the projections of the total angular momentum. When considering optical dipole transitions, not all transitions are allowed. The selection

9 Chapter 2. Concept of gray molasses cooling

Figure 2.3: Clebsch-Gordon coeﬃcients for a F = 1/2 → F ′ = 3/2 transition coupled by π light (full arrow), σ+ light (dotted arrow) and σ− light (dashed arrow).

Table 2.1: Selection rules for electric dipole transitions in atoms. Particular transitions are linked to respective necessary light polarization (Foot, 2005).

allowed transitions light polarization ∆S = 0 ∆L = ±1 ± ∆mL = 0, ±1 π, σ ∆J = 0, ±1 except J = 0 to J ′ = 0 ± ∆mJ = 0, ±1 π, σ ∆F = 0, ±1 except F = 0 to F ′ = 0 ± ∆mF = 0, ±1 π, σ rules are manifested in the Wigner-Eckart-Theorem and are shown in table 2.1. The

Clebsch-Gordon coefficient Ci,j for a transition from state i to a state j, describes the strength of this transition. Figure 2.3 shows the described system with Clebsch- Gordon coefficients for corresponding transitions. Light with π polarization drives transitions between levels with√ ∆mF = 0 being the vertical transition lines with 2 Clebsch-Gordon coefficients 3 . Transitions between levels with ∆mF = +1 and − + − + − ∆mF = 1 are driven by σ and σ light, respectively.√ Both σ and σ transitions 1 have two different Clebsch-Gordon coefficients 3 and 1. The Rabi frequency Ωi,j for a certain transition i → j in such a system is given by √ Ωi,j = Ci,jΓ I/2Isat. (2.9)

The light shift of the bright state |ψB⟩ in the three level system was given by (Ω2+Ω2) ∆E ∝ 1 2 . We can immediately deduce, that ∆E is proportional to the light δ1

10 2.3. Polarization gradient ﬁelds

σ σ σ σ σ σ (a) (b)

Figure 2.4: Polarization gradient fields in one dimension (a) and three dimensions (b). The resulting polarizations are constant in time but vary in space. (a) Two counter propagating beams in Lin⊥Lin configuration. Taken from (Dalibard and Cohen-Tannoudji, 1989). (b) Three counter propagating beams in σ+ and σ− configuration in all three spatial directions as in a magneto-optical trap configuration. intensity. But we also see, that the light shift depends on the light polarization, manifested in the Clebsch-Gordon coefficients. This means, even for spatially constant light intensity a varying polarization leads to spatially varying energy of the eigenstates. Several light field configurations, generating dark and bright states that are required for the gray molasses cooling, are possible. Two examples in one and three dimensions are shown in figure 2.4(a) and 2.4(b) respectively. The former shows the spatial dependence of the light polarization of two counter propagating light fields with linear polarization rotated by 90°. We call this configuration Lin⊥Lin. The intensity is constant along the z axis. At z = 0, light is linearly polarized changing to circular polarization at z = λ/8 and so forth. If we choose the quantization axis to be in z-direction the Lin⊥Lin configuration will yield alternating σ+ and σ− light with a period of λ/2. In figure 2.4(b) a light field configuration for a three dimensional polarization gradient field is shown. This configuration is also used in a MOT. Two counter propagating, circular polarized light beams in each spatial direction x, y, z respectively generate a three dimensional polarization gradient field. The one dimensional case is a simple configuration used in this thesis to analyze the effect of a polarization gradient field onto an atom. The three dimensional case was implemented into the experiment in the curse of this thesis.

11 Chapter 2. Concept of gray molasses cooling

2.4 Gray molasses of Lithium-6

In this section we introduce the energy level scheme of Lithium-6 and discuss new features of the gray molasses cooling due to the more complex level structure. 2 Figure 2.5 shows the full level scheme of the ground state 2 S1/2 and the ﬁrst two 2 2 excited states 2 P1/2 and 2 P3/2 of Lithium-6, where we use spectroscopic notation (2s+1) n LJ and n is the principle quantum number, s is the electron spin quantum number, L is the orbital angular momentum quantum number and J is the total 2 2 electronic angular momentum quantum number. Since the |2 S1/2⟩ → |2 P1/2⟩ transition is the transition to the ﬁrst excited state it is called the D1 line, which 2 2 has a transition frequency of 446.789 634 THz. The |2 S1/2⟩ → |2 P3/2⟩ transition is called the D2 line, and has a transition frequency of 446.799 677 THz.

2 Figure 2.5: Lithium-6 level scheme for the 2 S1/2 ground state and the ﬁrst two 2 2 excited states 2 P1/2 and 2 P3/2 including hyperﬁne splitting. Figure adapted from (Gehm, 2003).

The hyperfine splittings in the ground and the first excited states are δhfs = ′ 2 228.2 MHz and δhfs = 26.1 MHz, respectively. The hyperfine splitting of the 2 P3/2 ′′ × state is δhfs = 4.4 MHz. The linewidth of both excited states is Γ = 2π 5.87 MHz. 2 2 This shows, that the 2 P1/2 splitting is resolved but the splitting in 2 P3/2 is not.

The D1 and D2 lines of the system are split by ≈ 10 GHz. The states are split 2 ′ ′ 2 ′′ into F = 1/2, F = 3/2 for 2 S1/2, F = 1/2, F = 3/2 for 2 P1/2 and F = 1/2,

12 2.4. Gray molasses of Lithium-6

δ δ

Figure 2.6: Adaption of the three-level scheme for Lithium-6. Cooling and repumping frequencies couple both ground states to an excited state manifold with a blue detuning. The cooling beam detuning is δcool and the repumping beam detuning is δrep yielding ∆ = δcool − δrep.

′′ ′′ 2 F = 3/2 and F = 5/2 for the 2 P3/2 state. The line strength ratio between D1 and D2 line is approximately D2/D1 = 2 (Gehm, 2003).

The gray molasses cooling can be performed on the D1 line only, because the J = 1/2 → J ′ = 3/2 transition does not allow the process of gray molasses cooling. To further set the analogy between the Λ scheme and the Lithium-6 case we add two light frequencies to the system, coupling ground states |F = 1/2⟩ and |F = 3/2⟩ to the excited state |F ′ = 3/2⟩. This is shown in figure 2.6. In analogy to the MOT light we call the frequency of the transition |F = 3/2⟩ → |F ′ = 3/2⟩ the (0) | ⟩ → cooling frequency ωcool of the laser and the frequency of the transition F = 1/2 | ′ ⟩ (0) 1 F = 3/2 the repumping frequency ωrep of the laser . The difference between the (0) − (0) frequencies is given by the hyperfine splitting of the ground state ωrep ωcool = δhfs. − (0) − (0) We define δcool = ωcool ωcool and δrep = ωrep ωrep to be the detunings from the corresponding transitions and the relative detuning to be ∆ = δcool − δrep. Figure 2.7 shows the Lithium-6 transitions for cooling and repumping light be-

1It is important to note that this naming has traditional reasons and has nothing to do with the task the beams fulﬁll in the cooling process. We name them according to other publications (Burchianti et al., 2014; Rio Fernandes et al., 2012; Salomon et al., 2013)

13 Chapter 2. Concept of gray molasses cooling

σ π σ

2 2 Figure 2.7: Lithium-6 ground state 2 S1/2 and excited state 2 P1/2 under the in- ﬂuence of σ+ (left panel), π (middle panel) and σ− (right panel) light coupling ground and excited states resulting in a number of dark states marked by dashed circles. If a dashed circle contains two states, the dark state is a linear superposition of these states.

2 2 tween 2 S1/2 and 2 P1/2 for the diﬀerent light polarizations. Due to splitting of the hyperﬁne states into degenerate mF sub-states, dark states arise that have no analog in the Λ three level scheme. Table 2.2 shows all the dark and bright states of the system for σ+, π and σ− light polarization. If we consider the light polarizations σ+, a dark state appears which has no analogy in the Λ scheme. The |F = 3/2, mF = 3/2⟩ + state, for example, cannot couple to σ light since no transition with ∆mF = +1 − is available. The same counts for state |F = −3/2, mF = −3/2⟩ and light with σ polarization. The other dark states are linear superpositions of two states, one | {− 1 1 }⟩ | {− 1 1 }⟩ from state F = 1/2, mF = 2 , + 2 and one from F = 3/2, mF = 2 , 2 with a phase of π between them. The bright states of the system that are linear super- | {− 1 1 }⟩ positions have a phase shift of 0 between the states F = 1/2, mF = 2 , + 2 and | {− 1 1 }⟩ F = 3/2, mF = 2 , + 2 analogues to the bright state of the Λ scheme.

Under the conditions of two light frequencies coupling the ground state manifold 2 to the excited state manifold 2 P1/2 at detuning ∆ = 0 the system generates a dark state manifold and a bright state manifold with polarization dependent light shift. For the experimental setup we note, that we need two light ﬁelds with frequencies

ωcool = fF =3/2→F ′=3/2 + δcool (2.10)

ωrep = fF =1/2→F ′=3/2 + δrep (2.11) where fF =3/2→F ′=3/2 = 446.789 567 THz and fF =1/2→F ′=3/2 = 446.789 795 THz(Gehm, 2003).

14 2.5. Calculations for Lithium-6 in gray molasses

Table 2.2: Dark and bright states in the dressed state picture of Lithium-6 in a light field as shown in figure 2.6 at Raman condition for the three cases of purely σ+, π and σ− polarized light. The dark states are visualized in figure 2.7. Pol. Dark states Bright states | 3 3 ⟩ | 3 − 3 ⟩ ( F = 2 , mF = 2 ) ( F = 2 , mF = 2 ) σ+ √1 |F = 1 , m = 1 ⟩ − |F = 3 , m = 1 ⟩ √1 |F = 1 , m = 1 ⟩ + |F = 3 , m = 1 ⟩ (2 2 F 2 2 F 2 ) (2 2 F 2 2 F 2 ) √1 |F = 1 , m = − 1 ⟩ − |F = 3 , m = − 1 ⟩ √1 |F = 1 , m = − 1 ⟩ + |F = 3 , m = − 1 ⟩ 2 2 F 2 2 F 2 2 2 F 2 2 F 2 ( ) | 3 3 ⟩ F = 2 , mF = 2 √1 1 1 3 1 3 3 |F = , mF = ⟩ − |F = , mF = ⟩ |F = , m = − ⟩ π (2 2 2 2 2 ) ( 2 F 2 ) √1 |F = 1 , m = − 1 ⟩ − |F = 3 , m = − 1 ⟩ √1 |F = 1 , m = 1 ⟩ + |F = 3 , m = 1 ⟩ 2 2 F 2 2 F 2 (2 2 F 2 2 F 2 ) √1 |F = 1 , m = − 1 ⟩ + |F = 3 , m = − 1 ⟩ 2 2 F 2 2 F 2 | 3 − 3 ⟩ | 3 3 ⟩ ( F = 2 , mF = 2 ) ( F = 2 , mF = 2 ) − σ √1 |F = 1 , m = 1 ⟩ − |F = 3 , m = 1 ⟩ √1 |F = 1 , m = 1 ⟩ + |F = 3 , m = 1 ⟩ (2 2 F 2 2 F 2 ) (2 2 F 2 2 F 2 ) √1 |F = 1 , m = − 1 ⟩ − |F = 3 , m = − 1 ⟩ √1 |F = 1 , m = − 1 ⟩ + |F = 3 , m = − 1 ⟩ 2 2 F 2 2 F 2 2 2 F 2 2 F 2

2.5 Calculations for Lithium-6 in gray molasses

To obtain a deeper understanding of the process of gray molasses cooling, we take a closer look on the influence of the relative light detuning ∆, the magnetic field and the light phase. Therefore we numerically simulate the energy of a Lithium-6 atom inside a gray molasses field and discuss the cooling process. We solve the stationary Schrödinger equation of a Lithium-6 atom in a light field of two frequencies that couple ground states to the excited state manifold. We start by considering a one dimensional Lin⊥Lin configuration (as seen in figure 2.4(a)) and then consider a σ+ − σ− configuration in three dimensions (as shown in figure 2.4(b)). After we discuss light field polarizations, we construct the Hamilton operator describing the system. At the end of this section we discuss the results for the one dimensional, and three dimensional cases separately.

2.5.1 Description of light polarization

To simulate light-atom interaction we have to introduce several important char- acteristics of the light ﬁeld. Therefore, consider a monochromatic light wave with frequency ω which is connected to the wavenumber |⃗k| and wavelength λ |⃗| ω 2π ⃗ by k = c = λ , where the wavenumber k also describes the direction of light propagation. The full electric ﬁeld for light propagating in z direction, oscillating

15 Chapter 2. Concept of gray molasses cooling

in x direction with amplitude E0 and a phase ϕ can be written as

⃗ E(z, t) = Re[E0 exp(−i(kzz − ωt + ϕ))eˆx]. (2.12)

Here the light polarization is given by eˆx. A polarization turned by 90° will be 1 denoted eˆ . For circular polarization we write eˆ± = √ (eˆ ±ieˆ ), where ± accounts y 2 x y for right, and left polarization, respectively (Auzinsh et al., 2010). When adding two light fields, parts of the oscillating field enter into the polarization due to interference. In the following case we add two counter propagating beams in Lin⊥Lin configuration from figure 2.4(a). We obtain

⃗ E(z, t) = Re[E0 exp(−i(kzz − ωt + ϕ))eˆx] + Re[E0 exp(−i(−kzz − ωt))eˆy]. (2.13)

After separating the oscillating part, the polarization ⃗ϵp can be expressed as ( ( )) ( ( )) 2π ϕ 2π ϕ ⃗ϵ = exp i z + eˆ + exp −i z + eˆ . (2.14) p λ 2 x λ 2 y

This expression is consistent with the light polarization ﬁeld in ﬁgure 2.4(a). We { λ λ 3λ } can instantly see linear polarization at z = 0, 4 , 2 , 4 and left and right circular { λ 5λ } { 3λ 7λ } polarization at z = 8 , 8 and z = 8 , 8 , respectively.

Basis change. The polarization in the basis we used so far is not useful for a description of atom-light interaction. Instead, the π, σ± basis will yield a direct connection to atomic transitions as described in table 2.1 (Auzinsh et al., 2010). Because of this, we conduct a basis change of polarization using the following basis vectors 1 ϵˆ + = √ (eˆx + ieˆy) σ 2 1 (2.15) ϵˆ − = √ (eˆx − ieˆy) σ 2

ϵˆπ = eˆz where ϵˆπ, ϵˆσ+ and ϵˆσ− describe the respective light polarization that is required to drive transitions with ∆mF = 0, +1 and −1. Any general light polarization can now be written as a linear combination of these basis vectors.

⃗ϵ±(z) = ϵ+1ϵˆσ+ + ϵ0ϵˆπ + ϵ−1ϵˆσ− , (2.16)

16 2.5. Calculations for Lithium-6 in gray molasses where 2 2 2 ϵ+1 + ϵ0 + ϵ−1 = 1 (2.17) must be normalized. In the particular case of equation 2.14, the polarization can be written as: ( ) ( ) 4π 4π ⃗ϵ (z) = ϵˆ + cos z + ϵˆ − sin z , (2.18) p σ λ σ λ ( ) ( ) 4π 4π + where ϵ+1 = cos λ z and ϵ−1 = sin λ z . This describes alternating σ and σ− polarization. The discussed notation allows simple description of atom-light interaction. For example the Hamiltonian describing σ+ polarization light coupling to the atom we get

H = H0 + ϵσ+ V. (2.19)

Here H0 is the Hamiltonian without light interaction and V describes the coupling Hamiltonian. The same method can be extended to simulate the σ+ − σ− light ﬁeld in three dimension from ﬁgure 2.4(b) by adding 6 electromagnetic waves. The resulting polarization is then dependent on the location ⃗r and consists of π and σ± light.

2.5.2 Hamiltonian describing gray molasses

The full derivation of the Hamiltonian describing the Lithium-6 atom in a gray molasses has been done in (Gerken, 2014) and (Sievers et al., 2015). We only re- vise and explain the most important parts of the Hamiltonian construction. The 2 2 ground state manifold 2 S1/2 of Lithium-6 and the excited state manifold 2 P1/2 yield a total number of twelve states coupled by two light fields ωcool and ωrep as described in figure 2.6 and section 2.4. To construct the full Hamiltonian we assign its basis vectors to particular Lithium-6 states. Table 2.3 shows the assignment Af- ter performing the rotating wave approximation, the energy levels of the states are described only by the light detuning δcool, the relative detuning ∆ and the hyper- ′ fine splitting of the excited state δhfs = 26.1 MHz. The coupling of the ground to the excited states can be divided into two Hamiltonians HF =1/2 and HF =3/2, where

HF =1/2 describes the coupling due to the repumping light and HF =3/2 describes the coupling due to the cooling light. In total, we can write the Hamiltonian as

H = H0 + HF =1/2 + HF =3/2, (2.20)

17 Chapter 2. Concept of gray molasses cooling

Lithium-6 states E(F,mF ) RWA E(F,mF ) Abbreviated basis vectors

|F = 1/2, mF = −1/2⟩ -δhfs ∆ |1⟩ |F = 1/2, mF = 1/2⟩ -δhfs ∆ |2⟩ |F = 3/2, mF = −3/2⟩ 0 0 |3⟩ |F = 3/2, mF = −1/2⟩ 0 0 |4⟩ S-states |F = 3/2, mF = 1/2⟩ 0 0 |5⟩ |F = 3/2, mF = 3/2⟩ 0 0 |6⟩ ′ ′ ′ | ′ − ⟩ − | ⟩ F = 1/2, mF = 1/2 ω0 δhfs δcool + δhfs 7 ′ ′ ′ | ′ ⟩ − | ⟩ F = 1/2, mF = 1/2 ω0 δhfs δcool + δhfs 8 ′ |F = 3/2, mF ′ = −3/2⟩ ω0 δcool |9⟩ ′ |F = 3/2, mF ′ = −1/2⟩ ω0 δcool |10⟩

P-states ′ |F = 3/2, mF ′ = 1/2⟩ ω0 δcool |11⟩ ′ |F = 3/2, mF ′ = 3/2⟩ ω0 δcool |12⟩ Table 2.3: Lithium-6 states and respective energy in the normal basis and in the rotating wave approximation (RWA). The energy level is set to zero for the 2 2 |2 S1/2,F = 3/2⟩ states. ω0 is the energy between state |2 S1/2,F = 3/2⟩ 2 ′ and |2 P1/2,F = 3/2⟩. The energy in the RWA is obtained by rotat- 2 ing state |2 S1/2,F = 1/2⟩ by the repumper frequency ωrep and state 2 |2 S1/2,F = 3/2⟩ by the cooler frequency ωcool.

where H0 is the Hamiltonian of the Lithium-6 D1 manifold in the rotating wave approximation. The Hamiltonians can be written as

∑ H0 = |F = 1/2, mF ⟩ ~∆ ⟨F = 1/2, mF | + m ∑ ′ ′ |F , mF ′ ⟩ ~(δcool + δhfs,F ′ ) ⟨F , mF ′ | (2.21) ′ F ,mF ′ and ∑ ′ ± ~ ′ | ⟩⟨ ′ | HF =1 1/2 = ϵqΩF CF,mF ,F ,mF ′ ,q F, mF F , mF + H.c. (2.22) ′ mF ,F ,mF ′ ,q

F and F ′ denote the hyperﬁne states of the ground and excited state, respectively, ′ ′ and mF , mF ′ correspond to projections of F and F . δhfs,F ′ is the hyperﬁne splitting ′ ′ ′ of the excited state δhfs = 26.1 MHz for F = 1/2 and 0 for F = 3/2, q = + − {+1, 0, −1} represents the polarizations σ , π and σ , respectively, ΩF are the

′ Rabi frequencies depending on the intensity of the light ﬁeld and CF,mF ,F ,mF ′ ,q are the Clebsch-Gordon coeﬃcients for the corresponding transitions. Equation 2.23 shows the full Hamiltonian written out in detail separated into

18 2.5. Calculations for Lithium-6 in gray molasses

2 2 four sections . H1 describes the energy states of the 2 S1/2 ground state manifold. 2 H4 describes the energy states of the 2 P1/2 excited states manifold. H2 and H3 ′ describe the coupling between the ground and excited states. For shortening δcool = ′ δcool + δhfs. We diagonalize the Hamiltonian for different light field configurations. The resulting eigenvalues are the energy levels of the Lithium-6 atom in the given light field.   ~ H1 H2 H =   (2.23) 2 H3 H4

    ′ ∆ 0 0 0 0 0 δcool 0 0 0 0 0         ′   0 ∆ 0 0 0 0  0 δcool 0 0 0 0           0 0 0 0 0 0  0 0 δcool 0 0 0  H = 2 ×   H = 2 ×   1   4    0 0 0 0 0 0  0 0 0 δcool 0 0           0 0 0 0 0 0  0 0 0 0 δcool 0  0 0 0 0 0 0 0 0 0 0 0 δcool  √ √ √ √ √  1 1 2 4 2  ϵ0 √54 Ωr ϵ+ √27 Ωr ϵ− 9 Ωr ϵ0 √27 Ωr ϵ+√ 27 Ωr √0    − 1 − 1 2 4 2   ϵ− √27 Ωr ϵ0 54 Ωr √0 ϵ− 27 Ωr ϵ0 27 Ωr ϵ+ 9 Ωr    − 2 − 1 − 1   ϵ√+ 9 Ωc √0 ϵ0 6 Ωc ϵ√+ 3 Ωc √0 0  H =   2  4 − 2 − 1 − 1 − 4   ϵ0 Ωc ϵ+ Ωc ϵ− Ωc ϵ0 Ωc ϵ+ Ωc 0   √27 √ 27 3 √ 54 √ 27  − 2 4 4 1 − 1   ϵ− 27 Ωc ϵ0 √27 Ωc 0 ϵ− 27 Ωc ϵ0 54 Ωc ϵ√+ 3 Ωc − 2 1 1 0 ϵ− 9 Ωc 0 0 ϵ− 3 Ωc ϵ0 6 Ωc  √ √ √ √ √  1 − 1 − 2 4 − 2 ϵ0 √54 Ωr ϵ−√ 27 Ωr ϵ+ 9 Ωc ϵ0 √27 Ωc ϵ−√ 27 Ωc √0     1 − 1 − 2 4 − 2  ϵ+ √27 Ωr ϵ0 54 Ωr √0 ϵ+ 27 Ωc ϵ0 27 Ωc ϵ− 9 Ωc     2 − 1 1   ϵ−√ 9 Ωr √0 ϵ0 6 Ωc ϵ−√3 Ωc √0 0  H =   3  4 2 − 1 − 1 4  ϵ0 Ωr ϵ− Ωr ϵ+ Ωc ϵ0 Ωc ϵ− Ωc 0   √27 27 √3 √ 54 27   2 − 4 1 1  ϵ+ 27 Ωr √0 ϵ+ 27 Ωc ϵ0 54 Ωc ϵ− 3 Ωc √0  2 − 1 1 0 ϵ+ 9 Ωr 0 0 ϵ+ 3 Ωc ϵ0 6 Ωc

2 Note that the Rabi frequencies Ωc and Ωr do not include the Clebsch-Gordon coeﬃcients as shown in equation 2.9. Instead, the coeﬃcients are written individually.

19 Chapter 2. Concept of gray molasses cooling

Magnetic ﬁeld. Adding a magnetic ﬁeld to the Hamiltonian results in shifting

each energy level by the corresponding Zeeman shift ∆E(F,mF ). This can be calcu-

lated by the Breit-Rabi formula and adds ∆E(F,mF ) to each diagonal element of the matrix. The Breit-Rabi formula reads:

δ ∆E = − hfs,F − g µ m B± (F,mF ) 2(2I + 1) I B F √ 1 4m δ2 + F (g − g )µ Bδ + (g − g )2µ2 B2, (2.24) 2 hfs,F 2I + 1 J I B hfs,F J I B

where δhfs,F is the hyperﬁne splitting of the state, I is the nuclear spin, gI and gJ are the nuclear and electronic g-factors respectively, B is the magnetic ﬁeld, and

µB is the Bohr magneton (Foot, 2005). Figure 2.8(a) and 2.8(b) show the shifts for 2 2 the 2 S1/2 and 2 P1/2 states, respectively. These shifts are diﬀerent for every state and disturb the cooling process if on the order of the bright state light shift.

300

250

200

150

100

50 ft (MHz) ft i 0

-50 y Sh y g -100 er n

E -150

-200

-250

-300

0 20 40 60 80 100 120 140 160 Magnetic Field (G)

2 6 Figure 4: Magnetic-ﬁeld dependence(a) of the 2 S1/2 ground state of Li. (b)

2 2 Figure 2.8: Zeeman shift of the Lithium-6 2 S1/2 states (a) and 2 P1/2 states (b)

40 in a magnetic ﬁeld. Shifts are calculated using the Breit-Rabi formula. Figure taken from (Gehm, 2003). 30

10 ft (MHz) ft i ⊥ 2.5.3 Discussion0 of gray molasses in 1D Lin Lin ﬁeld y Sh y

g -10 er ⊥ In thisn section, we discuss the one dimensional cooling process in the Lin Lin E -20 configuration. Here, we numerically diagonalize the Hamiltonian that was described -30 in section 2.5.2 for three different detunings ∆ = {0, −0.25Γ, 0.5Γ}, a magnetic -40 field of B 0= 0.101 G 20and30 a phase40 50 shift60 of π/4 between cooling and repumping beam. Magnetic Field (G) −2 2 6 FigureFor 5: all Magnetic-field calculations dependence we of usethe 2 PI1sat/2 excited= 2 state.54 ofmWLi. cm as the saturation intensity of the

11 20 2.5. Calculations for Lithium-6 in gray molasses

D2 line of Lithium-6, Icool = 15Isat, Irep = 0.75Isat and δcool = 4Γ.

Figure 2.9: Light shift of the Lithium-6 ground state manifold over position z in a Lin⊥Lin gray molasses conﬁguration with Icool = 15Isat, Irep = 0.75Isat, −2 δcool = 4Γ and ∆ = 0, where Isat = 2.54 mW cm the saturation intensity of the D2 line. Black points show population in de dark state |ΨD⟩, light circles show population in the bright state |ΨB⟩. The black lines indicate cooling cycles.

We start by diagonalizing the Hamiltonian described in equation 2.23 for the Lin⊥Lin field for a relative detuning between cooling and repumping beam of ∆ = 0. The ground states of Lithium-6 in the gray molasses along the z axis in the dressed state picture are displayed in figure 2.9. As predicted in section 2.4, two dark states, red and blue lines in the figure, emerge with an energy which is almost constant along the spatial coordinate z. At points of fully circular polarized light with z = {λ/8, 3λ/8,...} an additional state revokes its light shift (violet and yellow lines in the figure, respectively). The green and the orange states are far detuned and take no part in the cooling mechanism. A clear structure, analogous to figure 2.1, can be seen. The black points show population in the dark state, the white circles show population in the bright state. A population evolution of one atom with velocity in the positive direction along the z axis is shown schematically. Beginning in the dark state the atom couples to the bright state at a point of low light shift of the bright state via velocity selective coupling. Then the atom climbs the potential hill, absorbs a photon near the top of the hill and decays back into the dark state. One cycle removes up to 1 MHz × ~ of kinetic energy from an atom. This corresponds to a temperature of 50 µK. When cooling from the MOT temperature of

21 Chapter 2. Concept of gray molasses cooling

240 µK down to the temperature of 40 µK we thus need around 4 cooling cycles. For one cooling cycle we have to account for three parts of the process:

• The velocity selective coupling process was described in equation 2.8 and is dependent on the light intensities, the detuning and the velocity of the atom. For the values introduced before, the transition probability is approximately 1%

• The climbing of the potential hill is only dependent on the velocity. For a temperature of 42 µK, the mean velocity of a Lithium-6 atom is approximately −1 λ 0.3 m s . It takes about 0.5 µs to travel a distance of 4 . 2π ≈ • The pumping process is on the order of the lifetime of the excited state Γ 170 ns. Compared to to the other durations this time is negligible.

Adding up these three parts we obtain a total time for a single cooling cycle of about 50 µs. For our previously discussed experimental parameters this yields a total cooling time of about 0.2 ms.

(a) (b)

Figure 2.10: Light shift of Lithium-6 ground state manifold over position z in a Lin⊥Lin gray molasses conﬁguration with Icool = 15Isat, Irep = 0.75Isat −2 and δcool = 4Γ, where Isat = 2.54 mW cm the saturation intensity of the D2 line. With: (a): ∆ = −0.25Γ and )b): ∆ = 0.5Γ.

Figure 2.10(a) shows the solution of the same Hamiltonian with the change of the detuning to ∆ = −0.25Γ. The two lowest states (blue and red lines in the ﬁgure) are shifted down by ≈ 1 MHz and are thus no longer dark states. Nevertheless, they have spatially constant energy and we therefore call them constant states. Due to the gap between the bright states and the constant states, the velocity selective

22 2.5. Calculations for Lithium-6 in gray molasses coupling is weakened in comparison to the case of ∆ = 0: atoms with a kinetic energy below 1 MHz × ~ can not couple to the bright state. Excluding atoms with an energy lower than 1 MHz×~ from the cooling process leads to a rise of the lower temperature limit by about 50 µK. Figure 2.10(b) shows the eigenstates for a detuning ∆ = 0.5Γ. The energy shift of the constant states (green and brown lines in the ﬁgure) exceeds the shift of the bright state. This leads to an inversion of the energy landscape where the constant states are energetically higher then the bright states. Here the coupling between constant and bright state is strongest at the maximal energy of the bright state. This causes an inversion of the process. The atom is accelerated down the potential hill leading to heating. For every cycle, kinetic energy of about 0.5 MHz×~ is added to the energy of the atom. These three cases show the sensibility of the cooling process on ∆. For ∆ > 0 and ∆ < 0 states with approximately constant light shift are generated that are leading to heating or suboptimal coolin conditions, respectively. Consequently, we expect the cooling to reach maximal eﬃciency at Raman condition that corresponds to ∆ = 0.

Figure 2.11: Light shift of the Lithium-6 ground state manifold with a homogeneous magnetic field B = 0.1 G over position z in a Lin⊥Lin gray molasses configuration with Icool = 15Isat, Irep = 0.75Isat and δcool = 4Γ, where −2 Isat = 2.54 mW cm the saturation intensity of the D2 line and ∆ = 0. For comparison with ∆ = 0 and B = 0 G see figure 2.9.

Another important aspect of the cooling process is the influence of an external magnetic field. In figure 2.11, we show a level structure of Lithium-6 in gray molasses under the influence of a homogeneous magnetic field of strength B = 0.1 G.

23 Chapter 2. Concept of gray molasses cooling

The Zeemann-effect causes a different energy shift for each state, leading to break down of the Raman condition. The energy structure does not show dark states or spatially constant states. The cooling process is not possible in this configuration. We deduce that the gray molasses cooling in this configuration is not possible for magnetic fields as small as 0.1 G.

Figure 2.12: Light shift of the Lithium-6 ground state manifold for a phase shift of π/4 between cooling and repumping beam over position z in a Lin⊥Lin gray molasses conﬁguration with Icool = 15Isat, Irep = 0.75Isat and −2 δcool = 4Γ, where Isat = 2.54 mW cm the saturation intensity of the D2 line and ∆ = 0.

In figure 2.12, a phase shift of ϕ = π/4 is set between the repumper and cooling beam. The energy landscape differs in the amplitude of the bright states compared the state of figure 2.9. The cooling process in this configuration works. This interesting case shows, that phase stability between repumper and cooling beam is not required, which considerably simplifies the experimental setup. With the calculations for the one dimensional gray molasses cooling, we found a high sensitivity of the cooling process towards the relative detuning ∆ and magnetic fields. High robustness of the cooling is found independent of the phases of the light fields.

2.5.4 Discussion of gray molasses in a 3D σ+-σ− ﬁeld

In this section we discuss the spatial evolution of the energy eigenstates for the Hamiltonian from equation 2.3 for the three dimensional σ+ − σ− light field configuration shown in figure 2.4(b).

24 2.5. Calculations for Lithium-6 in gray molasses

(a) (b)

Figure 2.13: Light shift of Lithium-6 for a light ﬁeld in 3D σ+ − σ− conﬁguration for a x and y unit cell, with Icool = 15Isat, Irep = 0.75Isat, δcool = 4Γ and ∆ = 0, at (a): z = 0λ, (b): z = 1/4λ, (c): z = 1/2λ and (d) z = 3/4λ. The state in blue color has almost no light shift on the x, y surface.

We focus on the case ∆ = 0 and B = 0. For this we consider characteristic slices of the unit cell x ∈ [0, λ], y ∈ [0, λ] and z ∈ {0, λ/4, λ/2, 3λ/4}. We do so, because in every z plane, the beam in z direction has a different phase and a different energy landscape arises. In figure 2.13(a) we see a cut through the unit cell at z = 0. Two states (the blue state and one state not visible because it is laying beneath the blue state) show no significant light shift over the x, y plane. The other states however show energy gradients in all directions, except at y = 0.5λ where we have no energy gradient along the x axis. This is called an escape channel where atoms could propagate without being cooled. The potential hill has a height of about 15 MHz which corresponds to a temperature of 750 µK. This augments the dissipation energy per cooling cycle such that an atom is cooled after just one cycle. The estimation for the cooling duration then reduces to 0.1 ms. Figures 2.13(b), 2.13(c) and 2.13(d) show the same configuration at z = 1/4λ, 1/2λ, and 3/4λ, respectively. At z = 1/4λ an escape channel at x = 0.5λ in y direction is visible. The escape channel without energy gradient from figure 2.13(a)

25 Chapter 2. Concept of gray molasses cooling

Figure 2.14: Light shift of Lithium-6 for a light field in a 3D σ+ −σ− configuration, at z = 0, for a x and y unit cell, with Icool = 15Isat, Irep = 0.75Isat, δcool = 4Γ and ∆ = 0 for a phase shift of ϕ = π/2 for the beam in positive x direction. and figure 2.13(b) are closed in all other planes in the unit cell. For a setup without phase stabilization the phase of a single beam will fluctuate on a time scale of 1 kHz. For an atom at 40 µK the atom has a velocity of approximately 0.3 m s−1. In the time of the order of the fluctuation the atom will thus propagate for about 300 µm. Fluctuations however can proceed on all six beams of the light field. This leads to a shortening of the time and the distance an atom can propagate on an escape channel. The effect of the channels can thus be neglected. Figure 2.14 shows the energy levels at z = 0 for the same configuration but a phase shift of ϕ = π/2 of the beam in positive x direction. The dark state, bright state structure is still preserved and the cooling process is possible. This supports the statement that the phase does not need to be stabilized for the cooling mechanism to work. In this chapter we explained the mechanism of gray molasses cooling. We described the effect of VSCPT into a dark state for atoms at rest for a Λ scheme and the resulting cooling in a polarization gradient field. We discussed the polarization dependent dark and bright states for Lithium-6 in a light field of two frequencies

ωcool and ωrep. Calculations on a one dimensional model showed a high sensitivity of the cooling process towards the relative detuning ∆ and magnetic ﬁelds and a high robustness towards phase shifts of beams. For a three dimensional case we estimated that one cooling cycle per atom cools the sample to the lowest temperature limit in a time of approximately 0.1 ms.

26 3 Implementation of gray molasses cooling of Lithium-6

In this chapter we describe the implementation and characterization of gray molasses cooling in our experiment. We start by discussing the complete experimental setup that ensures laser cooling starting from the laser system including frequency stabilization, the tapered amplifier (TA) system to three fibers for the x, y, and z beam, providing light for the experimental chamber table and finally the setup on the experimental chamber table itself. Then we introduce the experimental sample preparation. At the end of the chapter we discuss the measurements of the gray molasses cooling to characterize the final conditions of 3.2 × 107 atoms at a temperature of T = 42 µK. Measurements are done on the detuning ∆, the starting time after the MOT ∆t, the gray molasses duration τ, the cooling in magnetic offset and gradient field, the total intensity and finally the ratio between cooling and repumping intensity.

3.1 Setup of the gray molasses

The setup of the gray molasses cooling is divided into three separable parts. Figure 3.1 shows an overview with the key elements of the setup. The first part is the laser system consisting of the laser, the modulation transfer spectroscopy (MTS) for frequency stabilization and the TA system to amplify the laser power. This system provides the necessary light power at the right wavelength to seed system number two. The second part is the optical setup for cooling and repumping frequency preparation in which we generate the two necessary light frequencies, overlap them and couple the bichromatic beams into three different optical fibers. The fibers provide the light for the gray molasses on the experimental table which is the third and final part of the experimental setup.

27 Chapter 3. Implementation of gray molasses cooling of Lithium-6

Figure 3.1: Overview of the three major parts of the experimental setup. The laser system with MTS frequency stabilization and subsequent power amplification by a TA system. Frequency preparation for cooling and repumping beams and coupling into x, y and z optical fibers. Experimental chamber with gray molasses beams in MOT configuration.

3.1.1 Laser system and stabilization

The light source for the gray molasses cooling is a homebuilt diode laser in Littrow configuration. The light source is a HL6756MG laser diode from Thorlabs at ≈ 671 nm at a temperature of approximately 291 K. The output power is tunable from 2 mW to 20 mW. By tuning the laser current from the lasing threshold of 18 mA to 60 mA and the piezo voltage from 0 V to 150 V, the laser output frequency can be changed in a mode hop free range of more than 10 GHz. In figure 3.2, the required cooling and repumping frequencies we discussed in section 2.4 are shown in a Lithium-6 level diagram. Additionally, the laser frequency and the frequency inside the setup of the MTS are shown. The color coding used in the scheme is the same we use in figure 3.3. Each beam color stands for one particular frequency.

Modulation transfer spectroscopy. The laser is frequency stabilized by a MTS setup that has already been described in my bachelor thesis (Gerken, 2014). Ap- proximately 2 mW of the laser output power couples into the MTS setup which is depicted in figure 3.3 by the pink colored beam path. After passing an acousto- optical modulator (AOM) in double pass configuration, the frequency of the laser is shifted by twice the AOM frequency of 85 MHz in the positive first order. This

28 3.1. Setup of the gray molasses

δ δ s

Figure 3.2: Lithium-6 level scheme with coupling frequencies. The laser is detuned 2 by plus 170 MHz from the |F = 3/2⟩ → |2 P1/2⟩ transition frequency in an AOM double pass conﬁguration inside the spectroscopy setup. Cooling beam and repumping beam are then generated by AOM single passes shifting the frequencies by minus 114 MHz and plus 114 MHz−∆ from the laser frequency. results in a total shift of the laser frequency in the MTS of minus 170 MHz. The beam then enters the MTS setup which allows the locking of the laser frequency onto a particular Lithium-6 transition via a PID-feedback loop onto the piezo crystal voltage and the laser diode current (Müller, 2011). For this the beam traverses a cell of Lithium-6 vapor. A counter propagating beam obtains additional frequency sidebands after passing through an electro optical modulator (EOM). Both beams are overlapped inside the vapor cell. Due to four-wave-mixing, one of the frequency sidebands gets transfered onto the other beam depending on the frequency detuning from a Lithium-6 resonance. The resulting signal is registered on a photo-diode where the signal is demodulated and serves as an error signal. This gives the possibility to get an absolute frequency reference from a Lithium-6 transition for the laser. The hyperﬁne splitting of the excited state cannot be resolved by the spectroscopy due to a crossover peak between the |F = 3/2⟩ → |F ′ = 3/2⟩ and |F = 3/2⟩ → |F ′ = 1/2⟩ transitions. Taking this into account we lock the fre- 2 quency inside the MTS on the |F = 3/2⟩ → |2 P1/2⟩ transition of Lithium-6. This

29 Chapter 3. Implementation of gray molasses cooling of Lithium-6

2 yields a laser frequency of f|F =3/2⟩→|2 P1/2⟩ + 170 MHz. We approximate the line width of the laser by investigating the noise of the locking error signal to be around 1 MHz (Gerken, 2014).

Tapered amplifier system. The largest part of the laser power (around 15 mW) is used to seed a homebuilt tapered amplifier system designed in the group of S. Whitlock (Faraoni, 2014). The TA chip is a EYP-TPA-0670-00500-2003-CMT02- 0000 chip from Eagleyard for 670 nm. The TA is seeded after the light passes an aspherical lens from Thorlabs (C230TMD-B). The aspect ratio of 3:1 of the laser beam is advantageous for TA seeding. The diverging beam after the TA-chip is collimated by a similar aspherical lens from Thorlabs (Thorlabs C230TMD-B). After amplification the output power reaches 600 mW for a seed power of approximately 15 mW. The light passes a double stage optical isolator (Qioptiq FI-670-5TVC) to protect the TA from back reflections. After passing a cylindrical lens (Thorlabs LJ1267RM) the beam is collimated to a diameter of approximately 2.8 mm. Finally we have a linearly polarized single frequency beam with a power of around 450 mW.

3.1.2 Optical setup for light preparation

For the frequency preparation of the cooling and repumping beam, we need to detune the frequencies by minus 114 MHz and plus 114 MHz − ∆, respectively. We also have to account for the fact, that the cooler beam will have a much higher intensity of approximately Icool = 20Irep. For the cooling beam we use the minus first order of an AOM from Crystal Technology (3100-125) driven at a frequency of 114 MHz. The frequency of the zeroth order is again modified with an AOM by plus 114 MHz−∆ in order to generate the repumping beam. Both beams are overlapped on a non-polarizing 50:50 beam splitter cube since cooling and repumping beams need to have the same polarization. One bichromatic beam is coupled into the z fiber going to the z direction of the gray molasses in the experimental chamber. The other output of the beam splitter is again split by a non-polarizing 50:50 beam splitter cube. Both outputs are coupled into the x and y fibers. This yields a power 1 ratio between x, y and z beam of Px = Py = 2 Pz.A λ/2 plate and a polarizing beam splitter give the possibility to further change the ratio.

We now coupled light with two modes of frequencies fcool = f|F =3/2⟩→|F ′=3/2⟩+δcool and frep = f|F =1/2⟩→|F ′=3/2⟩ + δrep into the ﬁbers that transfer the light to the

30 3.1. Setup of the gray molasses vB v vBP vB P which lead directly to the P z and y , x vBB

PP vB /B P P v after the isolator. Frequencies are shifted by two AOMs to the cooling and v Δ P of gray molasses setup. Different colors correspond to different frequencies, as shown /BB P 450 mW 3.1 d . The laser is locked by modulation transfer spectroscopy. The laser seeds a tapered amplifier (TA) vB 3.2 : d d λ λ with an output power of Parts 1 and 2 fromin figure figure repumping frequencies. The light is coupled into the fibers by fiber couplers experimental chamber. : Figure 3.3

31 Chapter 3. Implementation of gray molasses cooling of Lithium-6

experimental chamber. This setup allows us to tune the frequency of δcool and

δrep = δcool − ∆ at the same time by changing the AOM frequency of the double pass in the spectroscopy. ∆ can be changed individually by tuning the frequency of the repumper AOM, while the power ratio Pcool/Prep is modiﬁed by altering the power of the radio frequency applied to the second AOM. The total power can be changed while keeping the Pcool/Prep ratio constant by changing the TA current.

3.1.3 Optical setup at experimental chamber

The setup for the beams on the experimental chamber table is different for beams in x, y and z direction. The beams in x and y direction are coupled out as can be seen in figure 3.4(a). The light is linearly polarized and sent through a λ/4 wave plate resulting in circular polarization. Then they are back-reflected on the other side of the chamber after passing another λ/4 wave plate twice. The angle between the x and y beams is approximately 90°.

The z beam is overlapped with the MOT beam in z direction on a non-polarizing 90:10 beam splitter (Thorlabs BSN16) as shown in figure 3.4(b). Thus, 10% of the MOT power and 90% of the power of the gray molasses beam in z direction is lost. This has to be tolerated due to spatial restrictions on the experimental chamber table. Like the MOT beam the gray molasses beam in z direction also passes a λ/4 wave plate and gets back-reflected after passing another λ/4 wave plate twice. The resulting beam and polarization configuration can be seen in figure 2.4(b).

Because of the 90:10 beam splitter, high power is required in the z beam. Since the intensity in all beams should be the same, we chose different beam sizes for each direction. The z beam is collimated by a 60FC-4-M12-07 coupler from Schaefter & Kirchhoff yielding a beam diameter of 2.15 mm. In x and y direction, we use 60FC-4-M20-07 fiber couplers from Schaefter & Kirchhoff with a beam diameter of 3.61 mm. Overall, we obtain the following powers for the beams before they enter the experimental chamber: Px,cool = 3.9 mW, Px,rep = 0.19 mW, Py,cool = 3.9 mW,

Py,rep = 0.19 mW, Pz,cool = 1.4 mW, Pz,rep = 0.1 mW. This yields intensities in every beam of approximately Icool = 15Isat and Irep = 0.75Isat, giving a ratio of 20 : 1 between the cooling and the repumping intensities.

32 3.2. Characterization of gray molasses cooling

(a) (b)

Figure 3.4: (a) Gray molasses beams (red) and MOT beams (blue) at experimental chamber. The cross in the middle indicates the beam in z direction. The beams in x, y and z direction are approximately perpendicular. (b) Beam path of beam in z direction on the experimental chamber table. The z beam is overlapped with the MOT beam on a 90:10 beam splitter. 10% of the MOT power and 90% of the gray molasses power are lost. Both beams have the same polarization. (Description of the optical elements can be found in ﬁgure 3.3.)

3.2 Characterization of gray molasses cooling

The characterization of the gray molasses cooling is at the heart of this thesis. It shows the results of the cooling mechanism as it is implemented in a standard experimental sequence. We reach temperatures of down to 42 µK for more than 80% of the MOT atoms augmenting the phase space density by a factor of ten.

3.2.1 Experimental sequence

The experimental sequence for the loading of the Lithium-6 MOT in the Mixtures experiment has been presented in several works (Pires, 2014; Ulmanis, 2015). The new implementation of gray molasses cooling however requires modifications to the original sequence. In figure 3.5, the sequence for the preparation of a Lithium-6 cloud is shown. The upper axis depicts the names of the stages. Time evolves to the right. On the left side, different components of the experiment are shown with lines indicating the mode of the elements. In the first phase, the MOT loading phase, we turn on the MOT coils, the MOT beams and the Zeeman slower for 1.5 s. At the end of this stage the Zeeman slower is turned off. In the next step

33 Chapter 3. Implementation of gray molasses cooling of Lithium-6

τ Δ

Figure 3.5: Sequence for the atom preparing of an ultracold Lithium-6 cloud. It starts with a MOT loading phase where the Zeeman slower coils and beam are on. After the MOT is loaded the cloud is transfered from the MOT coils to the curvature coils and finally compressed. After the compression phase the coils are turned off and gray molasses cooling is turned on after a waiting time ∆t for a variable time τ. Finally, the cloud is imaged via absorption imaging after a time of flight (TOF) t. we slowly turn of the MOT coils while turning on the curvature coils1 at the same time in 30 ms. In this process we do not loose any atoms. The next phase is the compression phase where the cloud is compressed by increasing the magnetic field gradient, lowering the MOT beam power and tuning the MOT frequencies closer to resonance in 70 ms. At this point we trap approximately 4 × 107 Lithium-6 atoms at a temperature of approximately 240 µK. We now turn off the MOT beams and curvature coils and implement the new cooling stage. After turning off the MOT beams and the curvature coils and a waiting time of ∆t we turn on the gray molasses cooling for a variable time τ. The high sensitivity of the gray molasses towards magnetic fields represented the biggest challenge in the Lithium-6 cooling approach. The magnetic field, generated by the MOT coils for the magnetic quadrupole field, needs approximately 10 ms to completely decay after switching the coils off (Repp, 2013). During this time the lithium cloud with a temperature of approximately 240 µK will have expanded.

1The name ’Curvature coils’ is used internally (Repp, 2013; Pires, 2014). Note that the curvature coils are small coils with few windings and a low inductance in anti Helmholtz conﬁguration, close to the atomic cloud sample.

34 3.2. Characterization of gray molasses cooling

When we considered turning the cooling phase on while the magnetic field is decaying, fluctuations of the field made gray molasses cooling to low temperatures impossible. Therefore we implemented an exchange phase in the sequence where the MOT cloud is transfered from the MOT coils into the magnetic field of the curvature coils. Since these coils are much closer to the chamber and much smaller, the magnetic field decays in less than 1 ms. Even though this time is too long to consider waiting for the field decay before turning on the gray molasses cooling phase, the curvature coils have smaller fluctuations arising through eddies. Since the atomic cloud is trapped in the center of the quadrupole field, where the field is close to B = 0 G, the largest part of the atoms can be cooled. However, cooling directly after turning off the magnetic field permits us to cool almost 80% of the atoms to the lowest possible temperature. In this thesis we focus on measurements of the gray molasses cooling. To characterize this process we take absorption images after the gray molasses cooling phase after a time of flight t. The basic principles of absorption imaging are explained in Ketterle et al. (1999). We use the imaging system to determine the cloud size and the number of atoms. By observing the cloud size after different time of flights we can determine the cloud temperature.

3.2.2 Measurements of gray molasses cooling

The measurements on the gray molasses cooling have been performed on an initial sample of approximately 4 × 107 Lithium-6 atoms at a temperature of 240 µK prepared by the method described in the last subsection. The sample is prepared and an absorption image is taken after a certain time of flight t. All measurement points of one measurement were taken in a random order. This method guarantees suppression of systematical errors developing over time, as for example thermal drifts on an AOM that can arise when one configuration is used several times in a row. For each point, we measure the atom number and the cloud size for 10 different time of flights. The number of atoms is deduced by averaging over the ten different measurements and the error is the standard error of the mean. The temperature is deduced by fitting the function √ 2 2 2 σ(t) = σ0 + σv t (3.1)

35 Chapter 3. Implementation of gray molasses cooling of Lithium-6

where σ0 is the size of the cloud at t = 0, σ(t) the size of the cloud at time t and σv the width of the Gaussian velocity distribution. This function describes the expansion of the cloud due to the Maxwell-Boltzmann distribution of the thermal atoms in the cloud. The temperature can be extracted from the ﬁt by the function

σ2m T = v , (3.2) kB where T is the temperature, m the atomic mass, σv is the velocity extracted from the fit and kB the Bolzmann constant. The error of the temperature determination is taken from the 68% -confidence interval of the fit. This interval is defined by the area that includes the next measurement point with a probability of 68%. By following this routine we obtain statistical errors for each of our measurement points independently.

For the following measurements the intensity of the beams Icool = 15Isat, Irep =

0.75Isat and the detuning δcool = 4Γ are kept constant if not noticed diﬀerent.

Raman detuning. In the ﬁrst measurement we apply gray molasses cooling with a duration of τ = 1 ms to the sample of atoms directly after tuning of the MOT

(∆t = 0). The detuning δcool = 4Γ was held constant, while the detuning between repumper and cooling frequency ∆ was tuned from −2.5Γ to 2Γ. The measurement is displayed in ﬁgure 3.6 where the left axis shows the temperature marked by black circles and the right side shows the number of atoms in blue squares. The Raman condition, ∆ = 0 is marked by a dashed vertical line. This plot shows four features:

• The ﬁrst feature emerges for large detunings from the Raman condition, in the range ∆ < −Γ or ∆ > Γ. The atom number is constant at around 2.5 × 107 atoms, being greater than 60% of the initial atom number. The temperature decreases to 150 µK. This is a decrease in temperature by a factor of approximately two compared to the MOT. In this conﬁguration the states are not separated into dark and bright states as discussed in section 2.4. The cooling process is perturbed.

• The second feature is found at ∆ ≈ 0.5Γ. Near this point the temperature rises and the atom number decreases down to the point where all atoms are lost. We simulated this situation with the calculations shown in ﬁgure 2.10 (b). Due to the detuning the dark states are energetically higher than the

36 3.2. Characterization of gray molasses cooling

4 , 5 3 5 0 T e m p e r a t u r e 4 , 0 N u m b e r o f A t o m s 3 0 0

3 , 5 ) 7 0 1

2 5 0 ( 3 , 0 s m ) o t

K 2 0 0 2 , 5 a µ

( f o T

1 5 0 2 , 0 r e b m

1 , 5 u 1 0 0 N 1 , 0 5 0 0 , 5 0 - 2 - 1 0 1 2 ∆ ( Γ) Figure 3.6: Temperature T (black circles) and number of atoms N (blue squares) after 1 ms of gray molasses cooling as a function of the relative detuning ∆ in units of Γ with δcool = 4Γ, Icool = 15Isat and Irep = 0.75Isat. The lowest temperature T = 42 µK is reached at the Raman condition ∆ = 0. The highest number of atoms is reached at ∆ ≈ −0.25Γ where we trap 100% of the atoms of the initial sample.

bright states. Since there is still optical pumping taking place, this results in heating. The observed atom loss is consistent with atoms escaping induced by heating. As our absorption imaging is not reliable for low number of atoms, there are missing points at around ∆ = 0.6Γ.

• The third feature is found at ∆ ≈ −0.25Γ. Here the temperature reaches T = 80 µK. The atom number reaches its maximum of 4 × 107 atoms. This is a capture eﬃciency of 100%. In ﬁgure 2.10 (a) we simulated this case. Dark states do not exist. Instead, the levels with spatially constant light shifts are shifted down by ≈ 7 MHz. As we described in equation 2.8, the coupling strength is dependent on the energy gap between both states. This means, that the velocity selective coupling is weakened compared to the case of ∆ = 0. The coupling process is therefore slown down explaining the higher temperature compared to ∆ = 0. In Sievers et al. (2015) it is pointed out, that feature two and three can be inversed when changing the repumper and

cooler intensity such that Irep/Icool = 20.

• The fourth and last feature shows the highest cooling eﬃciency. At ∆ = 0

37 Chapter 3. Implementation of gray molasses cooling of Lithium-6

our sample reaches the lowest temperature of 42 µK and an atom number of 3.2 × 107 atoms (an approximate efficiency of 80%). This is a temperature decrease by a factor of six while loosing approximately 20% of the atoms. Assuming constant volume this would result in an increase in phase space density by a factor of 25. However, the volume of the sample does not stay constant. During the first part of the cooling, where the magnetic field decays we observe an increase in the cloud size. This is due to the cooling mechanism not working in magnetic fields as shown in figure 2.11. This yields a lower gain in phase space density of an approximate factor of ten.

These four features have been observed by several groups in Lithium-6 (Burchianti et al., 2014; Sievers et al., 2015), Lithium-7 (Grier et al., 2013) and Potassium (Rio Fernandes et al., 2012; Salomon et al., 2013; Nath et al., 2013). For Lithium-6, temperatures around 42 µK have been reached by (Burchianti et al., 2014) and (Sievers et al., 2015). Our measurement is consistent with these publications and the simulations done in chapter 2.5. For the following measurements in this thesis we ﬁx the relative detuning to ∆ = 0.

Gray molasses timings. One further important parameter for gray molasses cooling are the timings, meaning the time between switching of the MOT and applying the gray molasses cooling ∆t and the duration τ. As discussed before, the high sensitivity of the gray molasses towards magnetic fields is challenging due to the slow decay of the magnetic field after the MOT is switched off. Due to this fact, we need to examine the efficiency of the gray molasses cooling depending on the waiting time after the MOT. Figure 3.7 shows the result of this measurement for an initial sample of 4 × 107 atoms at a temperature of 240 µK and a gray molasses duration of τ = 1 ms. We observe the atom number and the temperature for a waiting time ∆t from 0 ms to 1 ms. The temperature remains constant within the error bars. This means, that we reach the same temperature for our sample independent of the starting time and, more importantly, independent on the magnetic field decay. This means that either, the field decays much faster than expected or that the dependence of the decaying field has a weak effect on the cooling process. Due to the gradient field and the small size of the sample, we can assume that only the wings of the cloud are affected by the magnetic field. In the wings the density of atoms is much lower than in the middle of the cloud. Thus, as we will

38 3.2. Characterization of gray molasses cooling

6 0 3 , 4

T e m p e r a t u r e 3 , 2 N u m b e r o f A t o m s 5 0 3 , 0 ) 7 0

2 , 8 1 4 0 (

2 , 6 m ) o t K a µ

3 0 ( 2 , 4 f o T

2 , 2 e 2 0 b m 2 , 0 u N

1 0 1 , 8

1 , 6 0 0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0 T i m e a f t e r M O T ∆t ( m s ) Figure 3.7: Temperature T (black circles) and number of atoms N (blue squares) after τ = 1 ms of gray molasses cooling as a function of the time ∆t after MOT coils and light are turned off and the gray molasses beams are switched on. discuss later, the magnetic field gradient has a small effect on the sample. The number of atoms in this measurement decreases with increasing waiting time ∆t. This effect can be explained by considering the expansion of the cloud after the MOT is switched off. After some time, atoms will leave the area exposed by the gray molasses light. These atoms cannot be cooled and get lost. By extrapolating the number of atoms over ∆t, we approximate ∆t0 = 2.3 ms to be the time after which no atoms are cooled. An atom at 240 µK has a velocity of approximately 0.54 m s−1. For a gray molasses diameter of approximately 1.5 mm we can assume that all atoms leave the exposed area after a time of approximately 2.4 ms. The second timing of the gray molasses cooling we investigate is the duration τ of the molasses. Figure 3.8 shows the result for variable duration of the molasses. The molasses is turned on immediately after the MOT is switched off (∆t = 0) and 80% of the atoms are cooled. We see that the sample gets heated during the first 0.3 ms of the cooling phase. After this the temperature decreases rapidly. At 0.5 ms, the minimum is reached. These first 0.3 ms can be explained by the decaying magnetic field. The sample is heated up until the magnetic field falls below a critical value. Afterwards the sample is cooled rapidly. This might be another explanation for figure 3.7. The duration of the cooling time was long, which caused the sample get

39 Chapter 3. Implementation of gray molasses cooling of Lithium-6

5 0 0 T e m p e r a t u r e

4 0 0

) 3 0 0 K µ (

T 2 0 0

1 0 0

0 0 , 0 0 , 5 1 , 0 1 , 5 M o l a s s e s d u r a t i o n τ ( m s ) Figure 3.8: Temperature T as a function of molasses duration. heated in the ﬁrst part of the molasses duration and get cooled back down in the last part. The small bump between 0.5 ms and 1 ms could then be explained by an oscillation of the magnetic ﬁeld. No improvement in temperature is seen after 1 ms cooling time. In section 2.5, we approximated a cooling duration of 0.1 ms for a sample at 42 µK and a cooling of ∆200 µK. If we consider cooling from 440 µK to 42 µK, we still cool the atoms within one cycle. The approximated cooling time does not change. The observed cooling starts at 0.3 ms and ends after 0.5 ms yielding a total cooling time of twice the approximated length. In other experiments the molasses duration is set to 2 ms (Burchianti et al., 2014; Sievers et al., 2015), however in these experiments the number of atoms exceeds our sample size considerably, being 1 × 109 and 2 × 109 atoms respectively. At the FeLiKx experiment of Rudi Grimm cooling 1.2 × 108 atoms takes 1 ms (Fritsche, 2015) which is consistent with our observation.

Gray molasses in a magnetic field. An important aspect of gray molasses cooling is the fact that small magnetic fields can destroy the coherent Λ scheme, as discussed in section 2.5.3. In three measurements we examined the dependence of the gray molasses cooling for different magnetic field configurations. We start the measurement at Raman condition and study the atom number and

40 3.2. Characterization of gray molasses cooling temperature of the sample after τ = 1 ms of gray molasses cooling at ∆t = 0. During the gray molasses we apply a homogeneous and constant magnetic ﬁeld in x direction Bx.

5 0 0 5 0 0 ) )

T e m p e r a t u r e 3 . 5 7 T e m p e r a t u r e 3 . 5 7 N u m b e r o f A t o m s 0 N u m b e r o f A t o m s 0 4 0 0 1 4 0 0 1 ( (

3 . 0 s 3 . 0 s m m

) 3 0 0 ) 3 0 0 o o t t K K a a

µ 2 . 5 µ 2 . 5

( ( f f o o T T

2 0 0 2 0 0 r r

2 . 0 e 2 . 0 e b b

1 0 0 m 1 0 0 m u u

1 . 5 N 1 . 5 N 0 0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0

B X ( m G ) B Z ( m G ) (a) (b)

Figure 3.9: Temperature T (black circles) and number of atoms N (blue squares) for 1 ms long gray molasses cooling in a constant homogeneous magnetic ﬁeld in: (a) x direction and (b) z direction.

Figure 3.9(a) shows the result of this measurement. When increasing the magnetic field the temperature rises. Between 0 < Bx < 500 mG the temperature changes only slightly while the atom number increases. This could be due to the fact, that the gray molasses is proceeding in a small offset field due to power imbalance in the MOT beams. The field Bx we apply compensates this field. For

Bx > 500 mG the temperature increases dramatically while the atom number decreases until all atoms are lost at around Bx = 2000 mG. This is due to the fact we discussed in figure 2.11 where we showed that the whole energy structure breaks down for magnetic fields. Because we have now six beams instead of two, the energy shifts are stronger by a factor of six. The Zeeman shift ∆E = µBB attains the same order of magnitude as the energy shift of 20 MHz for a magnetic field of approximately 0.2 G. Figure 3.9(b) shows the same measurement as in figure 3.9(a) but with a magnetic

field Bz in z direction. Since our setup is completely symmetric, we do not expect large deviations from the previous measurement. The MOT field decaying in the initial phase of the cooling has a different gradient in x and z direction, nevertheless no influence on the measurement can be observed. One difference is the missing maximum in the atom number near Bz ≠ 0. This supports the assumption we made before that the cloud is not located at the zero field position in x direction.

41 Chapter 3. Implementation of gray molasses cooling of Lithium-6

9 0 4 , 0

T e m p e r a t u r e 8 0 N u m b e r o f A t o m s 3 , 5

7 0 3 , 0 ) 7 0 1

6 0 (

2 , 5 s m )

5 0 o t K a µ

2 , 0 ( f o T

4 0 r

1 , 5 e 3 0 b m u

1 , 0 N 2 0

1 0 0 , 5

0 0 , 0 0 5 1 0 1 5 2 0 2 5 3 0 M a g n e t i c F i e l d g r a d i e n t i n Z d i r e c t i o n ( G / c m ) Figure 3.10: Temperature T (black circles) and number of atoms N (blue squares) for 1 ms gray molasses cooling as a function of the magnetic quadrupole ﬁeld. The atomic cloud is situated in the center of the quadrupole ﬁeld.

Figure 3.10 shows the measurement of the dependence of gray molasses cooling on the MOT field left on with different gradient strengths. The magnetic field is described here as a gradient in z direction but it also consists of gradients in x and ∂B − ∂B − ∂B y directions ∂z = 2 ∂x = 2 ∂y . As indicated before, the cloud sample is located in the approximate center of the quadrupole field. This means, that only the wings of the cloud are perturbed by a considerable magnetic field. When assuming a cloud with a gaussian density distribution and a width of 1 mm in diameter, then the wings are affected by a field of 0.05 cm × ∇Bz. The wings of the cloud contain around 70% of the atoms. So, if for instance a gradient field of 5 G cm−1 is applied, then 70% of the atoms are affected by a field higher than 0.25 G. This explains why we do not see effects as strong as in figure 3.9(a) and 3.9(b). However, we still see an increase in temperature even at a small gradient field of 5 G cm−1. The loss of atoms can be explained by the heating in the wings of the sample. It is evident that cooling inside a magnetic quadrupole field is possible but the minimum temperature will not be as low as in zero magnetic quadrupole field. This assumption is supported by the results of other groups, where Pehoviak (2015) reached T = 25 µK after 10 ms of gray molasses cooling of Potassium-39 in a magnetic field and Salomon et al. (2013) reached 6 µK in zero magnetic field.

42 3.2. Characterization of gray molasses cooling

Light intensities. The last investigation for gray molasses cooling is the dependence of the cooling mechanism on the light intensities. Here we examine the dependence of the overall intensity while leaving the ratio between cooling and repumping intensities Icool/Isat = 20 constant. In the last measurement we probe the dependence on the ratio while keeping the cooler intensity Icool = 15Isat constant. Figure 3.11 shows the temperature and number of atoms after 1 ms of gray mo-

5 0 4 , 0

TT ee mm pp ee rr aa tt uu rr ee NN uu mm bb ee rr oo ff AA tt oo mm ss 3 , 5 4 0 )

3 , 0 7 0 1 (

2 , 5 s 3 0 m ) o t K a µ

2 , 0 ( f o T

2 0 r 1 , 5 e b m u

1 , 0 N 1 0 0 , 5

0 0 , 0 4 6 8 1 0 1 2 1 4 1 6 1 8

I c o o l / I s a t Figure 3.11: Temperature T (black circles) and number of atoms N (blue squares) for 1 ms long gray molasses cooling depending on intensity Icool while keeping Icool/Irep = 20 constant. lasses cooling applied at ∆t = 0 and for a relative detuning of ∆ = 0 as a function of the overall intensity. For an intensity of Icool < 4Isat we loose all atoms. For

Icool > 15Isat we cool the most atoms. The number of atoms increases with the intensity and converges towards a capture efficiency of approximately 80%. The temperature reaches its lowest point of 20 µK at an intensity of Icool = 5.5Isat for an atom number of 1.2 × 107. This can be explained by the excitation of off resonant hyperfine states which is proportional to the intensity and limits the temperature (Rio Fernandes et al., 2012). These results have also been observed by other groups (Salomon et al., 2013; Sievers et al., 2015). The higher capture rate at high intensities and the low temperatures at low intensities can be combined by a sequence of an initial cooling phase and a subsequent intensity ramp. This has been achieved for lithium and potassium, reaching temperatures lower by a factor of two compared to the initial cooling phase (Salomon et al., 2013; Sievers et al., 2015). This has not

43 Chapter 3. Implementation of gray molasses cooling of Lithium-6 been implemented in our experiment but is a further change in the experimental sequence that can be considered.

1 4 0 T e m p e r a t u r e 3 , 0 N u m b e r o f A t o m s )

1 2 0 7 2 , 5 0 1 (

1 0 0 m ) o

2 , 0 t K a µ

( f o T

8 0 r e

1 , 5 b m u

6 0 N 1 , 0

4 0 0 , 5 1 0 - 3 1 0 - 2 1 0 - 1

I r e p / I c o o l Figure 3.12: Temperature T (black circles) and number of atoms N (blue squares) for 1 ms of gray molasses cooling depending on repumping beam intensity while keeping Icool = 15Isat constant.

Figure 3.12 shows the measurement for a constant cooler intensity Icool = 15Isat and a changing ratio Irep/Icool. For a ratio higher than 0.03, no improvement in temperature and atom number can be seen. Lowering the ratio however drastically lowers the cooling effect and the amount of captured atoms. We assume that by lowering the repumper power under a certain threshold, coherent dark states cannot form. This could be observed by measuring the fluorescence of the cloud. If the atoms accumulate in the dark state, fluorescence should drop to zero. Presently this is not possible in our experiment. After the characterization of our gray molasses cooling we reached a temperature of 42 µK for approximately 80% of the initial atoms. The phase space density was increased by a factor of approximately ten. All of the observed features can be understood and explained here, and are consistent with the observations of other groups.

44 4 Conclusion and further developments

In this work we described the implementation of gray molasses cooling of Lithium- 6 in an ultracold gas experiment with Bose-Fermi mixture of Cesium-133 and Lithium-6 atoms. The newly implemented cooling process after a MOT phase enables to increase phase space density of the trapped atoms by a factor of ten, which represents a significant improvement in the starting conditions for subsequent loading of an ODT. In chapter 2 we discussed the general working scheme of gray molasses cooling. We started by introducing the three-level Λ scheme with a blue-detuned Raman- type coupling, yielding a dark and a bright state in the dressed state picture. We deduced motional coupling between the dark and the bright states, which is proportional to the bright state light shift and the velocity of the atom. Thus a moving atom, experiencing a spatially modulated light shift of the bright state, will undergo cooling due to repeated optical pumping from the bright state into the dark state until the atom is at rest. To describe the generation of the spatially-dependent light shift in an atom-light system we introduced a system with degenerate total angular momentum states. We showed that dipole transition rules lead to polarization- dependent light shifts. Two different polarization gradient fields in one and three dimensions were discussed which generated the spatially dependent light shift that was required for the cooling. We adapted these findings to the Lithium-6 sys- 2 tem where two frequencies coupling the ground state (2 S1/2) to the excited state 2 (2 P1/2) manifold, generate polarization dependent dark and bright states. We numerically calculated the energy levels of a Lithium-6 atom for a one and a three dimensional light polarization field configuration. There we observed that the system is sensitive towards the relative detuning of the beams and magnetic fields. We predicted optimal configuration at the Raman condition and at zero magnetic field. Changes due to phase shifts of the applied light fields did not disturb the cooling mechanism.

45 Chapter 4. Conclusion and further developments

In the first part of chapter 3 we described the setup for gray molasses generation including the laser system with frequency stabilization, tapered amplifier and the optical setup on the optical table and the experimental chamber. There we generate three orthogonal pairs of counter propagating laser beams in σ+ − σ− configuration, one in each Cartesian coordinate direction. The beam intensities are set to

Icool = 15Isat and Irep = 0.75Isat. The gray molasses cooling was implemented into the experimental sequence, where we apply a 1 ms gray molasses pulse directly after releasing the captured atoms from the MOT. In order to characterize the gray molasses cooling several experiments where performed. We measured temperature and atom number dependence on the Raman detuning ∆ and found the optimal setting at ∆ = 0. The timing of the cooling process was optimized to start immediately after the release from the MOT for a duration of 1 ms. Our measurements showed high sensitivity of the cooling process on homogeneous magnetic fields, where the temperature rose to 100 µK and 2.4 × 107 atoms at a magnetic field of 1 G. For a low light intensity of Icool = 6Isat we found even lower temperatures of approximately 20 µK for around 1 × 107 atoms. We determined that the atom number and temperature saturate for an intensity ratio between repumping and cooling beam of 1:20. For optimal parameters we measured a temperature of 42 µK. The sample contained 3.2 × 107 Lithium-6 atoms after 1 ms long gray molasses cooling. The atom number represents an 80% capture efficiency of the initial sample. These results show a decrease in the temperature of the MOT by a factor of six and a decrease in density by a factor of 0.7 yielding an approximate increase in phase space density by a factor of ten. Having successfully realized gray molasses cooling of Lithium-6 atoms, in the remaining part of this chapter we discuss further developments and perspectives for probing and simulation of few and many-body physics in ultracold lithium-cesium mixtures.

Loading into optical dipole trap. The loading of the ODT is a crucial part of the experimental setup. To reach the superﬂuid phase of Lithium-6 the phase space density of initially 7 × 10−5 (after gray molasses cooling) has to be increased by a factor of approximately 105. This is done by forced evaporative cooling in an ODT as described in Ketterle and van Druten (1996). Crucial to the performance of the evaporation process are the starting conditions of high number of atoms at low temperature. These parameters depend on the loading eﬃciency of the atom

46 cloud into the ODT. Due to the experimental upgrade of gray molasses cooling the initial conditions of the ODT loading changes. For the ODT we use a 200 W Yb-doped fiber laser (IPG YLR-200-LP-WC) at a wavelength of 1070 nm as described in Heck (2012). With the current setup we can trap Lithium-6 atoms with temperatures of up to 2 mK. To reach this trap depth the beams are focused onto the atomic cloud, leading to a trap −3 3 volume on the order of Vtrap ≈ 2×10 mm (Heck, 2012). The trap volume overlaps with approximately 2% of the initial MOT cloud volume, which is consistent with the loading efficiency of approximately 2% of the initial number of atoms. Inside the trap volume all atoms can be trapped due to the high trap depth compared to the cloud temperature. After the gray molasses cooling the cloud distribution in phase space changes significantly compared to the one of the MOT cloud. The temperature decreases by a factor of six while the density decreases by a factor of 0.7. The density decrease in the cloud will lead to a reduction in loading efficiency to approximately 1.3%. Figure 4.1 shows schematically the distribution in phase space of the cloud prepared in the MOT (red), after gray molasses cooling (orange) and the mode occupied by the ODT (blue). The overlap different distributions (not to scale) represent the loading efficiency of either the MOT cloud or gray molasses cloud into the ODT. To achieve higher loading efficiency, the phase-space overlap between the atom cloud and ODT distribution has to be optimized. We present two different approaches. First the waist of the ODT can be enlarged by using a 1:4 telescope, which will increase the trap volume by an approximate factor of 60 and reduce the trap depth to approximately 60 µK. This will yield an estimated loading efficiency of 60%. Due to the fixed optical setup this option is not flexible and decreases the efficiency of the forced evaporative cooling process, due to low trap frequencies. A more feasible method is a so-called time averaged potential. Here the spatial position of the ODT is modulated by an AOM with a frequency that is fast compared to the trap frequency, resulting in an increased volume and a reduced trap depth without impact onto the atoms (Grimm et al., 2000; Altmeyer et al., 2007; Lompe, 2008). The ODT volume, depth and loading efficiency would be similar to the first method, with the advantage of flexible trapping frequencies. Burchianti et al. (2014) used this method to load 2 × 107 atoms into an ODT from an initial sample of 1.2 × 109 Lithium-6 atoms. By additionally applying gray molasses cooling during the loading phase the temperature of the loaded cloud was decreased by

47 Chapter 4. Conclusion and further developments

Figure 4.1: Schematic representation of phase space distribution of atoms in the magneto-optical trap cloud (red), after gray molasses cooling (orange) and in the optical dipole trap (blue), where r is the spatial dimension and p is the momentum dimension (not in scale). Spread in momentum dimension is proportional to the square root of the temperature. a factor of two to 135 µK.

Sympathetic cooling in a Lithium-Cesium mixture. The enhanced Lithium-6 atom number inside the ODT will allow us to sympathetically cool Cesium-133 and Lithium-6 mixtures. In the case of two different species, two temperatures characterize the system as long as both species are not in thermal contact. When two gases with different initial temperature thermalize, one gas is heated while the other gas is cooled. For gases trapped in different potentials it is possible to evaporatively cool one gas, while maintaining the other and keeping both gases in thermal contact. This leads to sympathetic cooling of the trapped atoms. This is shown schematically in figure 4.2 for atoms of type A in red and atoms of type B in blue. Atoms A are evaporatively cooled while thermalizing with atoms B. Losses occur only on type A atoms while atoms of type B are cooled sympathetically. This method has been used to sympathetically cool Lithium-7 with Cesium-133 (Mudrich et al., 2002). Another possibility is the sympathetic cooling of cesium with lithium. Furthermore, this approach is routinely applied in many groups to realize a degenerate Fermi gas (Mewes et al., 1999; Zwierlein et al., 2006a; Ketterle and Zwierlein, 2008).

48 Figure 4.2: Scheme of sympathetic cooling. Two diﬀerent types of atoms A (red) and B (blue) are trapped in diﬀerent potentials. Type-A atoms are cooled via forced evaporative cooling. Type-B atoms are in thermal contact with type-A atoms and thus sympathetically cooled. Figure taken from (Taglieber, 2008).

The conditions in the Cesium-133 and Lithium-6 system are favorable for sympathetic cooling. Due to two broad Feshbach resonances (∆ ≈ 60 G) the scattering length between cesium and lithium can be tuned to optimize thermalization rates (Repp et al., 2013; Tung et al., 2013; Pires et al., 2014; Ulmanis et al., 2015). Additionally the cloud overlap between lithium and cesium can be optimized by bichromatic species selective trapping potentials (Häfner, 2013; Ulmanis, 2015; Ul- manis et al., 2016). Sympathetic cooling will yield ﬂexible preparation sequences and enable to produce large samples of ultracold Lithium-6 and Cesium-133 mixtures.

Cesium-133 as a thermometer of a degenerate Lithium-6 Fermi gas. Since Bose-Einstein condensates and strongly interacting Fermi gases do not follow free ballistic expansion nor are their atom velocities normal distributed, a relation between the cloud expansion and temperature, as normally used in ultracold atom thermometry, cannot be easily applied to determine sample temperatures. Instead alternative methods have to be used. For example, for the temperature determination of a BEC, the fraction of atoms in the ground state, given by the relation

3 Nth = N(T/TC ) , (4.1) can be used as a thermometer. Here, N is the total number of atoms in the cloud,

Nth is the thermal part of the cloud, T is the temperature and TC is the critical ~ω¯ 1/3 temperature given by TC = 0.94 N inside harmonic trap with average trap kB 1/3 frequency ω¯ = (ωxωyωz) . By ﬁtting a bimodal distribution to the cloud shape, the condensate fraction and thus the temperature can be extracted (Ketterle et al.,

49 Chapter 4. Conclusion and further developments

1999). In the case of a superfluid Fermi gas, the temperature can be extracted by a bimodal function fitted to the cloud shape. However the characteristic change in the shape of the cloud due to superfluidity can easily be misinterpreted (Ketterle and Zwierlein, 2008) and its precise determination requires a well characterized high- quality imaging system. Since temperature measurements play a critical role in the study of ultracold Bose-Fermi mixtures, the capability of precise thermometry will be decisive for our experiment. The challenges in a temperature measurement of highly degenerate Fermi gases can be partially overcome in a mixed-species setup. The two component experiment allows to use Cesium-133 atoms as a thermometer of Lithium-6. Determining the temperature of a small cesium cloud in thermal contact with a lithium cloud leads to knowledge about the lithium temperature with minimal impact on the sample. The temperature determination of the Cesium-133 atoms can either by done by TOF for a thermal cloud at temperatures exceeding 40 µK, or by fitting the bimodal distribution to the Cesium-133 BEC as described before at temperatures below 40 µK. This simple method will lead to reliable temperature determinations of the degenerate Fermi sample.

Double superfluid Bose-Fermi mixture of Cesium-133 and Lithium-6. With the realization of a superfluid Lithium-6 sample due to the experimental upgrade presented in this thesis, the preparation of a double superfluid Bose-Fermi mixture of Cesium-133 and Lithium-6 will be possible. For a long time, theories have tried to understand and model superfluid Bose-Fermi mixtures. The hard sphere model predicts a phase separation of the two components in the mixture of Helium-3 and

Helium-4 below a critical temperature TC (Cohen and Leeuwen, 1961). Landau’s Fermi liquid theory (Bardeen et al., 1957) gave the first model for a critical temperature of such a transition. In the mean-field regime with different interaction strength (scattering length) other phases are predicted. For example stable phases are predicted for small Bose-Fermi scattering lengths and collapse of the double superfluid phase is predicted for negative Bose-Fermi scattering length (Ospelkaus and Ospelkaus, 2008). Experimental studies of such systems are still scarce. Only three double Bose- Fermi superfluids have been realized until today. Ferrier-Barbut et al. (2014) used a mixture of Lithium-7 and Lithium-6, Roy et al. (2016) realized a mixture of Ytterbium-174 and Lithium-6, and Yao et al. (2016) used Potassium-41 and

50 Lithium-6. An ultracold mixture of Cesium-133 and Lithium-6 shows favorable properties for the probing of double Bose-Fermi superfluidity. The high mass imbalance is predicted to change the interaction energies between the superfluids (Zhang et al., 2014; Enss and Zwerger, 2009). The two broad magnetic Feshbach resonances at 843 G and 889 G are ideal to explore the phase diagram of the mass-imbalanced double superfluid of not only weak interacting (mean-field) Bose-Fermi mixtures, but also strongly interacting regimes. Furthermore, different phases are predicted for varying relative particle numbers of the mixed gas. Fluctuations in excitations in the superfluid Fermi sea and BEC are predicted to alter these phase boundaries.

Polaron physics in ultracold Bose-Fermi mixtures. The investigation of non- equilibrium dynamics of a single impurity interacting with a many-body quantum system is important for the understanding for many physical phenomena. Proto- typical examples are the coupling of an electron to phonons inside a solid, known as the Fröhlich polaron (Fröhlich, 1954; Tempere et al., 2009; Scelle et al., 2013), or coupling of massive particles to the Higgs field. The study of such Bose polarons, an impurity immersed in a bosonic bath, has generated a lot of interest (Scelle et al., 2013; Blinova et al., 2013; Rath and Schmidt, 2013). Only in 2016 the first Bose polarons were realized in ultracold atom experiments by Jørgensen et al. (2016) and Hu et al. (2016). Another important system in our understanding of the fundamental structure of quantum matter is the one of an impurity immersed in a degenerate Fermi sea. Such systems have been extensively studied in ultracold atom experiments (Schirotzek et al., 2009; Kohstall et al., 2012; Koschorreck et al., 2012). The ground and excited state energies, the effective mass, and the quasi particle residue of such Fermi polarons have been observed. Recently real time quench dynamics, the evolution of an impurity suddenly brought into the interacting state by applying a radio frequency pulse, have been studied by Cetina et al. (2016). An ultracold mixture of Cesium-133 and Lithium-6 is well suited for the probing of Bose polarons and Fermi polarons. The bosonic and fermionic nature of the mixture components give the possibility to create Bose polarons, by preparing Lithium-6 impurities in a Cesium-133 BEC and the realization of Fermi polarons, by probing Cesium-133 impurities in a degenerate Fermi sea of Lithium-6. The large mass imbalance between Cesium-133 and Lithium-6 allows to access the physically

51 Chapter 4. Conclusion and further developments relevant limit of a static polaron, where the mass of the infinitely heavy impurity drastically simplifies the theoretic description. The high control over the intra and inter species scattering length using magnetic Feshbach resonances, will enable tuning of, not only impurity-bath interactions between the bath atoms. This con- trollability will permit to explore different polaron flavors, such as, Landau-Pekar or bubble polarons (Blinova et al., 2013). Concluding we can say that the Mixtures experiment is looking towards an excit- ing future. The gray molasses cooling opens the door to many experimental possi- bilities with the promise to approach new physics in strongly interacting quantum matter.

52 Bibliography

Altmeyer, A., Riedl, S., Wright, M. J., Kohstall, C., Denschlag, J. H., and Grimm, R. (2007). Dynamics of a strongly interacting Fermi gas: The radial quadrupole mode. Phys. Rev. A, 76:033610.

Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Cornell, E. A. (1995). Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269(5221):198–201.

Arimondo, E. (1992). Laser manipulation of atoms and ions, proceedings of the summer school of physics E. Fermi, Varenna, 1991. Vol. CXVIII, (North-Holland, Amsterdam).

Auzinsh, M., Budker, D., and Rochester, S. M. (2010). Optically polarized atoms: Understanding light-atom interactions. Oxford University Press.

Aymar, M. and Dulieu, O. (2005). Calculation of accurate permanent dipole mo- ments of the lowest 1,3Σ+ states of heteronuclear alkali dimers using extended basis sets. The Journal of Chemical Physics, 122(20):204302.

Bardeen, J., Cooper, L. N., and Schrieﬀer, J. R. (1957). Theory of superconductivity. Phys. Rev., 108:1175–1204.

Blinova, A. A., Boshier, M. G., and Timmermans, E. (2013). Two polaron ﬂavors of the Bose-Einstein condensate impurity. Phys. Rev. A, 88:053610.

Bloch, I., Dalibard, J., and Zwerger, W. (2008). Many-body physics with ultracold gases. Rev. Mod. Phys., 80:885–964.

Bradley, C. C., Sackett, C. A., Tollett, J. J., and Hulet, R. G. (1995). Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett., 75:1687–1690.

53 Bibliography

Burchianti, A., Valtolina, G., Seman, J. A., Pace, E., De Pas, M., Inguscio, M., Zaccanti, M., and Roati, G. (2014). Eﬃcient all-optical production of large 6Li

quantum gases using D1 gray-molasses cooling. Phys. Rev. A, 90:043408.

Castin, Y., Dalibard, J., and Cohen-Tannoudji, C. (1991). The limits of Sisyphus cooling. ETS Editrice Pisa.

Cetina, Y., Jag, M., Lous, R. S., Fritsche, I., Walraven, J. T. M., Grimm, R., Levinsen, J., Parish, M. M., Schmidt, R., Knap, M., and Demler, E. (2016). Ultrafast many-body interferometry of impurities coupled to a Fermi sea. arXiv:1604.07423.

Chin, C., Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Denschlag, J. H., and Grimm, R. (2004). Observation of the pairing gap in a strongly interacting Fermi gas. Science, 305(5687):1128–1130.

Chu, S. (1998). Nobel lecture: The manipulation of neutral particles. Rev. Mod. Phys., 70:685–706.

Cohen, E. and Leeuwen, J. V. (1961). On supermobility in an isotopic mixture of a hard sphere Bose and Fermi gas at T = 0. Physica, 27:1157–1160.

Cohen-Tannoudji, C. N. (1998). Nobel lecture: Manipulating atoms with photons. Rev. Mod. Phys., 70:707–719.

Cornell, E. A. and Wieman, C. E. (2002). Nobel lecture: Bose-Einstein condensation in a dilute gas, the ﬁrst 70 years and some recent experiments. Rev. Mod. Phys., 74:875–893.

Courteille, P., Freeland, R. S., Heinzen, D. J., van Abeelen, F. A., and Verhaar, B. J. (1998). Observation of a Feshbach resonance in cold atom scattering. Phys. Rev. Lett., 81:69–72.

Dalibard, J. and Cohen-Tannoudji, C. (1989). Laser cooling below the doppler limit by polarization gradients: Simple theoretical models. J. Opt. Soc. Am. B, 6(11):2023–2045.

Davis, K. B., Mewes, M. O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. (1995). Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett., 75:3969–3973.

54 Bibliography

DeMarco, B. and Jin, D. S. (1999). Onset of fermi degeneracy in a trapped atomic gas. Science, 285(5434):1703–1706.

Duarte, P. M., Hart, R. A., Hitchcock, J. M., Corcovilos, T. A., Yang, T.-L., Reed, A., and Hulet, R. G. (2011). All-optical production of a lithium quantum gas using narrow-line laser cooling. Phys. Rev. A, 84:061406.

Edwards, D. M. (2013). A smooth polaron-molecule crossover in a Fermi system. IOP Publishing J. Phys.: Condens. Matter, 25(425602).

Enss, T. and Zwerger, W. (2009). Superﬂuidity near phase separation in Bose-Fermi mixtures. The European Physical Journal B, 68(3):383–389.

Esslinger, T., Sander, F., Weidemüller, M., Hemmerich, A., and Hänsch, T. W. (1996). Subrecoil laser cooling with adiabatic transfer. Phys. Rev. Lett., 76:2432– 2435.

Faraoni, G. (2014). Nonlinear light propagation through a strongly interacting Rydberg gas. Master thesis, Universita di Pisa Facolt di Scienze Matematiche, Fisiche e Naturali.

Ferrier-Barbut, I., Delehaye, M., Laurent, S., Grier, A. T., Pierce, M., Rem, B. S., Chevy, F., and Salomon, C. (2014). A mixture of Bose and Fermi superﬂuids. Science, 345(6200):1035.

Foot, C. J. (2005). Atomic Physics. Oxford University Press.

Fritsche, I. (2015). Sub-Doppler cooling of fermionic lithium. Master thesis, Uni- versität Innsbruck.

Fröhlich, H. (1954). Electrons in lattice ﬁelds. Advances in Physics, 3(11):325–361.

G. Grynberg, J.-Y. C. (1994). Proposal for a magneto-optical lattice for trapping atoms in nearly-dark states. Europhys. Lett. 27 41.

Gehm, M. E. (2003). Properties of 6Li. available online at www.physics.ncsu.edu/jet/techdocs/pdf/PropertiesOfLi.pdf.

Gerken, M. (2014). Modulationtransferspektroskopie in 6Li zur Laserkühlung mit grauer Melasse. Bachelor thesis, Ruprecht-Karls-Universität Heidelberg, Ger- many.

55 Bibliography

Greiner, M., Regal, C. A., and Jin, D. S. (2003). Emergence of a molecular Bose- Einstein condensate from a Fermi gas. Nature, 426(6966):537–540.

Grier, A. T., Ferrier-Barbut, I., Rem, B. S., Delehaye, M., Khaykovich, L., Chevy, F., and Salomon, C. (2013). λ-enhanced sub-Doppler cooling of lithium atoms

in D1 gray molasses. Phys. Rev. A, 87:063411.

Grimm, R., Weidemüller, M., and Ovchinnikov, Y. B. (2000). Optical dipole traps for neutral atoms. Advances In Atomic, Molecular, and Optical Physics, 42:95 – 170.

Hadzibabic, Z., Stan, C. A., Dieckmann, K., Gupta, S., Zwierlein, M. W., Görlitz, A., and Ketterle, W. (2002). Two-species mixture of quantum degenerate Bose and Fermi gases. Phys. Rev. Lett., 88:160401.

Häfner, S. (2013). A tunable optical dipole trap for 6Li and 133Cs. Master’s thesis, Ruperto-Carola-University of Heidelberg.

Hänsch, T. W. and Schawlow, A. L. (1975). Cooling of gases by laser radiation. Optics Communications, 13:68–69.

Heck, R. (2012). All-optical formation of an ultracold gas of fermionic lithium close to quantum degeneracy. Master’s thesis, Ruperto-Carola-University of Heidel- berg.

Hu, M.-G., Van de Graaﬀ, M., Kedar, D., Corson, J. P., Cornell, E. A., and Jin, D. S. (2016). Bose polarons in the strongly interacting regime. arXiv:1605.00729.

Inouye, S., Andrews, M. R., Stenger, J., Miesner, H.-J., Stamper-Kurn, D. M., and Ketterle, W. (1998). Observation of Feshbach resonances in a Bose-Einstein condensate. Nature, 392(6672):151–154.

Jochim, S., Bartenstein, M., Altmeyer, A., Hendl, G., Riedl, S., Chin, C., Hecker Denschlag, J., and Grimm, R. (2003). Bose-Einstein condensation of molecules. Science, 302(5653):2101–2103.

Jørgensen, N. B., Wacker, L., Skalmstang, K. T., Parish, M. M., Levinsen, J., Christensen, R. S., Bruun, G. M., and Arlt, J. J. (2016). Observation of attractive and repulsive Polarons in a Bose-Einstein condensate. Phys. Rev. Lett., 117:055302.

56 Bibliography

Ketterle, W. (2002). Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser. Rev. Mod. Phys., 74:1131–1151.

Ketterle, W., Durfee, D. S., and Stamper-Kurn, D. M. (1999). Making, probing and understanding Bose-Einstein condensates. In M. Inguscio, S. S. and Wieman, C., editors, Proceedings of the International School of Physics "Enrico Fermi", Course CXL, pages 67–176. IOS Press, Amsterdam.

Ketterle, W. and van Druten, N. J. (1996). Evaporative cooling of trapped atoms. Advances in Atomic, Molecular, and Optical Physics, 37:181.

Ketterle, W. and Zwierlein, M. W. (2008). Making, probing and understanding ultracold fermi gases. In Inguscio, M., Ketterle, W., , and Salomon, C., editors, Proceedings of the International School of Physics Enrico Fermi, volume Course CLXIV, page 95. IOS Press, Amsterdam.

Kohstall, C., Zaccanti, M., Jag, M., Trenkwalder, A., Massignan, P., Bruun, G. M., Schreck, F., and Grimm, R. (2012). Metastability and coherence of repulsive polarons in a strongly interacting Fermi mixture. Nature, 485(7400):615–618.

Koschorreck, M., Pertot, D., Vogt, E., Fröhlich, B., Feld, M., and Köhl, M. (2012). Attractive and repulsive Fermi polarons in two dimensions. Nature, 485(7400):619–622.

Kraemer, T., Herbig, J., Mark, M., Weber, T., Chin, C., Nägerl, H.-C., and Grimm, R. (2004). Optimized production of a cesium Bose-Einstein condensate. Applied Physics B: Lasers and Optics.

Lett, P. D., Watts, R. N., Westbrook, C. I., Phillips, W. D., Gould, P. L., and Metcalf, H. J. (1988). Observation of atoms laser cooled below the Doppler limit. Phys. Rev. Lett., 61:169–172.

Lompe, T. (2008). An apparatus for the production of molecular Bose-Einstein condensates. Master’s thesis, Ruprecht-Karls-Universität Heidelberg, Germany.

McKay, D. C., Jervis, D., Fine, D. J., Simpson-Porco, J. W., Edge, G. J. A., and Thywissen, J. H. (2011). Low-temperature high-density magneto-optical trapping of potassium using the open 4S → 5P transition at 405 nm. Phys. Rev. A, 84:063420.

57 Bibliography

Mewes, M.-O., Ferrari, G., Schreck, F., Sinatra, A., and Salomon, C. (1999). Si- multaneous magneto-optical trapping of two lithium isotopes. Phys. Rev. A, 61:011403.

Mudrich, M., Kraft, S., Singer, K., Grimm, R., Mosk, A., and Weidemüller, M. (2002). Sympathetic cooling with two atomic species in an optical trap. Phys. Rev. Lett., 88:253001.

Müller, R. (2011). Towards an ultracold mixture of 6Li and 133Cs. Master thesis, Ruprecht-Karls-Universität Heidelberg, Germany.

Nath, D., Easwaran, K., Rajalakshmi, G., and Unnikrishnan, C. S. (2013). Quantum-interference-enhanced deep sub-Doppler cooling of 39K atoms in gray molasses. Physical review A, 88:053407–1–7.

Ospelkaus, C. and Ospelkaus, S. (2008). Heteronuclear quantum gas mixtures. Journal of Physics B: Atomic, Molecular and Optical Physics, 41(20):203001.

Papoﬀ, F., Mauri, F., and Arimondo, E. (1992). Transient velocity-selective coherent population trapping in one dimension. Journal of the Optical Society of America B, 9:321–331.

Pehoviak, N. (2015). Strongly interacting ultracold potassium atoms via Rydberg dressing. Master thesis, Ruprecht-Karls-Universität Heidelberg, Germany.

Phillips, W. D. (1998). Nobel lecture: Laser cooling and trapping of neutral atoms. Rev. Mod. Phys., 70:721–741.

Pires, R. (2014). Eﬁmov resonances in an ultracold mixture with extreme mass imbalance. PhD thesis, Ruperto-Carola-University of Heidelberg.

Pires, R., Repp, M., Ulmanis, J., Kuhnle, E. D., Weidemüller, M., Tiecke, T. G., Greene, C. H., Ruzic, B. P., Bohn, J. L., and Tiemann, E. (2014). Analyzing Feshbach resonances: A 6Li-133Cs case study. Phys. Rev. A, 90:012710.

Pupillo, G., Micheli, A., Büchler, H., and Zoller, P. (2008). Cold molecules: Cre- ation and applications. Taylor & Francis.

Quéméner, G. and Julienne, P. S. (2012). Ultracold molecules under control! Chem- ical Reviews, 112(9):4949–5011.

58 Bibliography

Rath, S. P. and Schmidt, R. (2013). Field-theoretical study of the Bose polaron. Phys. Rev. A, 88:053632.

Repp, M. (2013). Interspecies Feshbach resonances in an ultracold, optically trapped Bose-Fermi mixture of cesium and lithium. PhD thesis, Ruperto-Carola- University of Heidelberg.

Repp, M., Pires, R., Ulmanis, J., Heck, R., Kuhnle, E. D., Weidemüller, M., and Tiemann, E. (2013). Observation of interspecies 6Li-133Cs Feshbach resonances. Phys. Rev. A, 87:010701.

Rio Fernandes, D., Sievers, F., Kretzschmar, N., Wu, S., Salomon, C., and Chevy, F. (2012). Sub-Doppler laser cooling of fermionic 40K atoms in three-dimensional gray optical molasses. EPL, 100:63001.

Roy, R., Green, A., Bowler, R., and Gupta, S. (2016). Two-element mixture of Bose and Fermi superﬂuids. arXiv:1607.03221.

Salomon, C., Shlyapnikov, G., and Cugliandolo, L. (2013). Many-body physics with ultracold gases: Lecture notes of the Les Houches summer school: Volume 94, July 2010. Lecture Notes of the Les Houches Summer School. OUP Oxford.

Scelle, R., Rentrop, T., Trautmann, A., Schuster, T., and Oberthaler, M. K. (2013). Motional coherence of fermions immersed in a Bose gas. Phys. Rev. Lett., 111:070401.

Schirotzek, A., Wu, C.-H., Sommer, A., and Zwierlein, M. W. (2009). Observation of Fermi Polarons in a tunable Fermi liquid of ultracold atoms. Phys. Rev. Lett., 102:230402.

Schreck, F., Khaykovich, L., Corwin, K. L., Ferrari, G., Bourdel, T., Cubizolles, J., and Salomon, C. (2001). Quasipure Bose-Einstein condensate immersed in a Fermi sea. Phys. Rev. Lett., 87:080403.

Schunck, C. H., Shin, Y.-i., Schirotzek, A., and Ketterle, W. (2008). Determi- nation of the fermion pair size in a resonantly interacting superﬂuid. Nature, 454(7205):739–743.

Sidorenkov, L. A., Tey, M. K., Grimm, R., Hou, Y.-H., Pitaevskii, L., and Stringari, S. (2013). Second sound and the superﬂuid fraction in a Fermi gas with resonant interactions. Nature, 498(7452):78–81.

59 Bibliography

Sievers, F., Kretzschmar, N., Rio Fernandes, D., Suchet, D., Rabinovic, M., Wu, S., Parker, C., Khaykovich, L., Salomon, C., and Chevy, F. (2015). Simultaneous 6 40 sub-Doppler laser cooling of fermionic Li and K on the D1 line: Theory and Experiment. Physical review A, 91:023426–1–11.

Taglieber, M. (2008). Quantum degeneracy in an atomic Fermi-Fermi-Bose mixture. PhD thesis, Ludwig-Maximilians-Universität München.

Tempere, J., Casteels, W., Oberthaler, M. K., Knoop, S., Timmermans, E., and Devreese, J. T. (2009). Feynman path-integral treatment of the BEC-impurity polaron. Phys. Rev. B, 80:184504.

Truscott, A. G., Strecker, K. E., McAlexander, W. I., Partridge, G. B., and Hulet, R. G. (2001). Observation of Fermi pressure in a gas of trapped atoms. Science, 291(5513):2570–2572.

Tung, S.-K., Parker, C., Johansen, J., Chin, C., Wang, Y., and Julienne, P. S. (2013). Ultracold mixtures of atomic 6Li and 133Cs with tunable interactions. Phys. Rev. A, 87:010702.

Ulmanis, J. (2015). Universality and non-universality in the heteronuclear Eﬁmov scenario with large mass imbalance. PhD thesis, Ruperto-Carola-University of Heidelberg.

Ulmanis, J., Häfner, S., Pires, R., Kuhnle, E. D., Weidemüller, M., and Tiemann, E. (2015). Universality of weakly bound dimers and Eﬁmov trimers close to Li-Cs Feshbach resonances. New Journal of Physics, 17(5):055009.

Ulmanis, J., Häfner, S., Pires, R., Werner, F., Petrov, D. S., Kuhnle, E. D., and Weidemüller, M. (2016). Universal three-body recombination and Eﬁmov resonances in an ultracold Li-Cs mixture. Phys. Rev. A, 93:022707.

Weber, T., Herbig, J., Mark, M., Nägerl, H.-C., and Grimm, R. (2003). Bose- Einstein condensation of cesium. Science, 299(5604):232–235.

Weidemüller, M., Esslinger, T., Ol’shanii, M. A., Hemmerich, A., and Hänsch, T. W. (1994). A novel scheme for eﬃcient cooling below the photon recoil limit. Europhysics Letters, 27(2):109.

60 Bibliography

Weidemüller, M. and Zimmermann, C. (2003). Interactions in ultracold gases: From atoms to molecules. Wiley-VCH.

Yao, X.-C., Chen, H.-Z., Wu, Y.-P., Liu, X.-P., Wang, X.-Q., Jiang, X., Deng, Y., Chen, Y.-A., and Pan, J.-W. (2016). Observation of two-species vortex lattices in a mixture of mass-imbalance Bose and Fermi superﬂuids. arXiv:1606.01717.

Zhang, R., Zhang, W., Zhai, H., and Zhang, P. (2014). Calibration of the interaction energy between Bose and Fermi superﬂuids. Phys. Rev. A, 90:063614.

Zwierlein, M. W., Schirotzek, A., Schunck, C. H., and Ketterle, W. (2006a). Fermionic superﬂuidity with imbalanced spin populations. Science, 311(5760):492–496.

Zwierlein, M. W., Schunck, C. H., Schirotzek, A., and Ketterle, W. (2006b). Direct observation of the superﬂuid phase transition in ultracold Fermi gases. Nature, 442(7098):54–58.

Zwierlein, M. W., Stan, C. A., Schunck, C. H., Raupach, S. M. F., Gupta, S., Hadz- ibabic, Z., and Ketterle, W. (2003). Observation of Bose-Einstein condensation of molecules. Phys. Rev. Lett., 91:250401.

Acknowledgments

At this point I would like to thank everyone who was involved or supported me in the termination of this thesis.

Special thanks go to. . .

. . . my supervisor Prof. Dr. Matthias Weidemüller for trust, motivation and inspira- tion during the past two years of my studies.

. . . Dr. Juris Ulmanis and Stephan Häfner, for the tremendous support and tutoring during the past two years and especially the last few months.

. . . the group of Quantum Dynamics of Atomic and Molecular Systems and to Clau- dia Krämer, for creating this highly enjoyable working environment.

. . . Prof. Dr. Selim Jochim and Dr. Shannon Whitlock and the groups of Ultracold Quantum Gases and Exotic Quantum Matter, for fruitful discussions and good times with "mulled wine with a rum-soaked sugarloaf lit above it" and "kicker".

. . . my parents Rosa and Christoph, for the deepest support one can wish for.

. . . Frédéric and Elena, for inﬂuence beyond their knowledge.

. . . Rosemarie, for everything.

. . . my Heidelberg crew and the "1337" crew for reminding me of life outside the lab.

. . . the University library for love and hate, joy and disgust. Bye bye......

Erklärung:

Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.

Heidelberg, den 24. August 2016 ......